[GOAL]
α : Type u_1
β : Type u_2
X Y Z : Pointed
f : Pointed.Hom X Y
g : Pointed.Hom Y Z
⊢ (g.toFun ∘ f.toFun) X.point = Z.point
[PROOFSTEP]
rw [Function.comp_apply, f.map_point, g.map_point]
[GOAL]
α : Type u_1
β : Type u_2
X : Type ?u.6897
Y : Pointed
f : typeToPointed.obj X ⟶ Y
⊢ (fun f =>
        { toFun := fun o => Option.elim o Y.point f,
          map_point :=
            (_ :
              (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
      ((fun f => f.toFun ∘ some) f) =
    f
[PROOFSTEP]
apply Pointed.Hom.ext
[GOAL]
case toFun
α : Type u_1
β : Type u_2
X : Type ?u.6897
Y : Pointed
f : typeToPointed.obj X ⟶ Y
⊢ ((fun f =>
          { toFun := fun o => Option.elim o Y.point f,
            map_point :=
              (_ :
                (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                  (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
        ((fun f => f.toFun ∘ some) f)).toFun =
    f.toFun
[PROOFSTEP]
funext x
[GOAL]
case toFun.h
α : Type u_1
β : Type u_2
X : Type ?u.6897
Y : Pointed
f : typeToPointed.obj X ⟶ Y
x : (typeToPointed.obj X).X
⊢ Pointed.Hom.toFun
      ((fun f =>
          { toFun := fun o => Option.elim o Y.point f,
            map_point :=
              (_ :
                (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                  (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
        ((fun f => f.toFun ∘ some) f))
      x =
    Pointed.Hom.toFun f x
[PROOFSTEP]
cases x
[GOAL]
case toFun.h.none
α : Type u_1
β : Type u_2
X : Type ?u.6897
Y : Pointed
f : typeToPointed.obj X ⟶ Y
⊢ Pointed.Hom.toFun
      ((fun f =>
          { toFun := fun o => Option.elim o Y.point f,
            map_point :=
              (_ :
                (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                  (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
        ((fun f => f.toFun ∘ some) f))
      none =
    Pointed.Hom.toFun f none
[PROOFSTEP]
exact f.map_point.symm
[GOAL]
case toFun.h.some
α : Type u_1
β : Type u_2
X : Type ?u.6897
Y : Pointed
f : typeToPointed.obj X ⟶ Y
val✝ : X
⊢ Pointed.Hom.toFun
      ((fun f =>
          { toFun := fun o => Option.elim o Y.point f,
            map_point :=
              (_ :
                (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                  (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
        ((fun f => f.toFun ∘ some) f))
      (some val✝) =
    Pointed.Hom.toFun f (some val✝)
[PROOFSTEP]
rfl
[GOAL]
α : Type u_1
β : Type u_2
X'✝ X✝ : Type ?u.6897
Y✝ : Pointed
f : X'✝ ⟶ X✝
g : X✝ ⟶ (forget Pointed).obj Y✝
⊢ ↑((fun X Y =>
              { toFun := fun f => f.toFun ∘ some,
                invFun := fun f =>
                  { toFun := fun o => Option.elim o Y.point f,
                    map_point :=
                      (_ :
                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                          (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) },
                left_inv :=
                  (_ :
                    ∀ (f : typeToPointed.obj X ⟶ Y),
                      (fun f =>
                            { toFun := fun o => Option.elim o Y.point f,
                              map_point :=
                                (_ :
                                  (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                    (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                          ((fun f => f.toFun ∘ some) f) =
                        f),
                right_inv :=
                  (_ :
                    ∀ (f : X ⟶ (forget Pointed).obj Y),
                      (fun f => f.toFun ∘ some)
                          ((fun f =>
                              { toFun := fun o => Option.elim o Y.point f,
                                map_point :=
                                  (_ :
                                    (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                            f) =
                        f) })
            X'✝ Y✝).symm
      (f ≫ g) =
    typeToPointed.map f ≫
      ↑((fun X Y =>
                { toFun := fun f => f.toFun ∘ some,
                  invFun := fun f =>
                    { toFun := fun o => Option.elim o Y.point f,
                      map_point :=
                        (_ :
                          (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                            (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) },
                  left_inv :=
                    (_ :
                      ∀ (f : typeToPointed.obj X ⟶ Y),
                        (fun f =>
                              { toFun := fun o => Option.elim o Y.point f,
                                map_point :=
                                  (_ :
                                    (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                            ((fun f => f.toFun ∘ some) f) =
                          f),
                  right_inv :=
                    (_ :
                      ∀ (f : X ⟶ (forget Pointed).obj Y),
                        (fun f => f.toFun ∘ some)
                            ((fun f =>
                                { toFun := fun o => Option.elim o Y.point f,
                                  map_point :=
                                    (_ :
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                              f) =
                          f) })
              X✝ Y✝).symm
        g
[PROOFSTEP]
apply Pointed.Hom.ext
[GOAL]
case toFun
α : Type u_1
β : Type u_2
X'✝ X✝ : Type ?u.6897
Y✝ : Pointed
f : X'✝ ⟶ X✝
g : X✝ ⟶ (forget Pointed).obj Y✝
⊢ (↑((fun X Y =>
                { toFun := fun f => f.toFun ∘ some,
                  invFun := fun f =>
                    { toFun := fun o => Option.elim o Y.point f,
                      map_point :=
                        (_ :
                          (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                            (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) },
                  left_inv :=
                    (_ :
                      ∀ (f : typeToPointed.obj X ⟶ Y),
                        (fun f =>
                              { toFun := fun o => Option.elim o Y.point f,
                                map_point :=
                                  (_ :
                                    (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                            ((fun f => f.toFun ∘ some) f) =
                          f),
                  right_inv :=
                    (_ :
                      ∀ (f : X ⟶ (forget Pointed).obj Y),
                        (fun f => f.toFun ∘ some)
                            ((fun f =>
                                { toFun := fun o => Option.elim o Y.point f,
                                  map_point :=
                                    (_ :
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                              f) =
                          f) })
              X'✝ Y✝).symm
        (f ≫ g)).toFun =
    (typeToPointed.map f ≫
        ↑((fun X Y =>
                  { toFun := fun f => f.toFun ∘ some,
                    invFun := fun f =>
                      { toFun := fun o => Option.elim o Y.point f,
                        map_point :=
                          (_ :
                            (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                              (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) },
                    left_inv :=
                      (_ :
                        ∀ (f : typeToPointed.obj X ⟶ Y),
                          (fun f =>
                                { toFun := fun o => Option.elim o Y.point f,
                                  map_point :=
                                    (_ :
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                              ((fun f => f.toFun ∘ some) f) =
                            f),
                    right_inv :=
                      (_ :
                        ∀ (f : X ⟶ (forget Pointed).obj Y),
                          (fun f => f.toFun ∘ some)
                              ((fun f =>
                                  { toFun := fun o => Option.elim o Y.point f,
                                    map_point :=
                                      (_ :
                                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                          (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                                f) =
                            f) })
                X✝ Y✝).symm
          g).toFun
[PROOFSTEP]
funext x
[GOAL]
case toFun.h
α : Type u_1
β : Type u_2
X'✝ X✝ : Type ?u.6897
Y✝ : Pointed
f : X'✝ ⟶ X✝
g : X✝ ⟶ (forget Pointed).obj Y✝
x : (typeToPointed.obj X'✝).X
⊢ Pointed.Hom.toFun
      (↑((fun X Y =>
                { toFun := fun f => f.toFun ∘ some,
                  invFun := fun f =>
                    { toFun := fun o => Option.elim o Y.point f,
                      map_point :=
                        (_ :
                          (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                            (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) },
                  left_inv :=
                    (_ :
                      ∀ (f : typeToPointed.obj X ⟶ Y),
                        (fun f =>
                              { toFun := fun o => Option.elim o Y.point f,
                                map_point :=
                                  (_ :
                                    (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                            ((fun f => f.toFun ∘ some) f) =
                          f),
                  right_inv :=
                    (_ :
                      ∀ (f : X ⟶ (forget Pointed).obj Y),
                        (fun f => f.toFun ∘ some)
                            ((fun f =>
                                { toFun := fun o => Option.elim o Y.point f,
                                  map_point :=
                                    (_ :
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                              f) =
                          f) })
              X'✝ Y✝).symm
        (f ≫ g))
      x =
    Pointed.Hom.toFun
      (typeToPointed.map f ≫
        ↑((fun X Y =>
                  { toFun := fun f => f.toFun ∘ some,
                    invFun := fun f =>
                      { toFun := fun o => Option.elim o Y.point f,
                        map_point :=
                          (_ :
                            (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                              (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) },
                    left_inv :=
                      (_ :
                        ∀ (f : typeToPointed.obj X ⟶ Y),
                          (fun f =>
                                { toFun := fun o => Option.elim o Y.point f,
                                  map_point :=
                                    (_ :
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                              ((fun f => f.toFun ∘ some) f) =
                            f),
                    right_inv :=
                      (_ :
                        ∀ (f : X ⟶ (forget Pointed).obj Y),
                          (fun f => f.toFun ∘ some)
                              ((fun f =>
                                  { toFun := fun o => Option.elim o Y.point f,
                                    map_point :=
                                      (_ :
                                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                          (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                                f) =
                            f) })
                X✝ Y✝).symm
          g)
      x
[PROOFSTEP]
cases x
[GOAL]
case toFun.h.none
α : Type u_1
β : Type u_2
X'✝ X✝ : Type ?u.6897
Y✝ : Pointed
f : X'✝ ⟶ X✝
g : X✝ ⟶ (forget Pointed).obj Y✝
⊢ Pointed.Hom.toFun
      (↑((fun X Y =>
                { toFun := fun f => f.toFun ∘ some,
                  invFun := fun f =>
                    { toFun := fun o => Option.elim o Y.point f,
                      map_point :=
                        (_ :
                          (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                            (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) },
                  left_inv :=
                    (_ :
                      ∀ (f : typeToPointed.obj X ⟶ Y),
                        (fun f =>
                              { toFun := fun o => Option.elim o Y.point f,
                                map_point :=
                                  (_ :
                                    (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                            ((fun f => f.toFun ∘ some) f) =
                          f),
                  right_inv :=
                    (_ :
                      ∀ (f : X ⟶ (forget Pointed).obj Y),
                        (fun f => f.toFun ∘ some)
                            ((fun f =>
                                { toFun := fun o => Option.elim o Y.point f,
                                  map_point :=
                                    (_ :
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                              f) =
                          f) })
              X'✝ Y✝).symm
        (f ≫ g))
      none =
    Pointed.Hom.toFun
      (typeToPointed.map f ≫
        ↑((fun X Y =>
                  { toFun := fun f => f.toFun ∘ some,
                    invFun := fun f =>
                      { toFun := fun o => Option.elim o Y.point f,
                        map_point :=
                          (_ :
                            (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                              (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) },
                    left_inv :=
                      (_ :
                        ∀ (f : typeToPointed.obj X ⟶ Y),
                          (fun f =>
                                { toFun := fun o => Option.elim o Y.point f,
                                  map_point :=
                                    (_ :
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                              ((fun f => f.toFun ∘ some) f) =
                            f),
                    right_inv :=
                      (_ :
                        ∀ (f : X ⟶ (forget Pointed).obj Y),
                          (fun f => f.toFun ∘ some)
                              ((fun f =>
                                  { toFun := fun o => Option.elim o Y.point f,
                                    map_point :=
                                      (_ :
                                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                          (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                                f) =
                            f) })
                X✝ Y✝).symm
          g)
      none
[PROOFSTEP]
rfl
[GOAL]
case toFun.h.some
α : Type u_1
β : Type u_2
X'✝ X✝ : Type ?u.6897
Y✝ : Pointed
f : X'✝ ⟶ X✝
g : X✝ ⟶ (forget Pointed).obj Y✝
val✝ : X'✝
⊢ Pointed.Hom.toFun
      (↑((fun X Y =>
                { toFun := fun f => f.toFun ∘ some,
                  invFun := fun f =>
                    { toFun := fun o => Option.elim o Y.point f,
                      map_point :=
                        (_ :
                          (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                            (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) },
                  left_inv :=
                    (_ :
                      ∀ (f : typeToPointed.obj X ⟶ Y),
                        (fun f =>
                              { toFun := fun o => Option.elim o Y.point f,
                                map_point :=
                                  (_ :
                                    (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                            ((fun f => f.toFun ∘ some) f) =
                          f),
                  right_inv :=
                    (_ :
                      ∀ (f : X ⟶ (forget Pointed).obj Y),
                        (fun f => f.toFun ∘ some)
                            ((fun f =>
                                { toFun := fun o => Option.elim o Y.point f,
                                  map_point :=
                                    (_ :
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                              f) =
                          f) })
              X'✝ Y✝).symm
        (f ≫ g))
      (some val✝) =
    Pointed.Hom.toFun
      (typeToPointed.map f ≫
        ↑((fun X Y =>
                  { toFun := fun f => f.toFun ∘ some,
                    invFun := fun f =>
                      { toFun := fun o => Option.elim o Y.point f,
                        map_point :=
                          (_ :
                            (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                              (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) },
                    left_inv :=
                      (_ :
                        ∀ (f : typeToPointed.obj X ⟶ Y),
                          (fun f =>
                                { toFun := fun o => Option.elim o Y.point f,
                                  map_point :=
                                    (_ :
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                              ((fun f => f.toFun ∘ some) f) =
                            f),
                    right_inv :=
                      (_ :
                        ∀ (f : X ⟶ (forget Pointed).obj Y),
                          (fun f => f.toFun ∘ some)
                              ((fun f =>
                                  { toFun := fun o => Option.elim o Y.point f,
                                    map_point :=
                                      (_ :
                                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                          (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                                f) =
                            f) })
                X✝ Y✝).symm
          g)
      (some val✝)
[PROOFSTEP]
rfl


[GOAL]
α : Type u
f g : MonoOver α
t : f ⟶ g
⊢ (fun f => Set.range f.obj.hom) f ≤ (fun f => Set.range f.obj.hom) g
[PROOFSTEP]
rintro a ⟨x, rfl⟩
[GOAL]
case intro
α : Type u
f g : MonoOver α
t : f ⟶ g
x : (𝟭 (Type u)).obj f.obj.left
⊢ Comma.hom f.obj x ∈ (fun f => Set.range f.obj.hom) g
[PROOFSTEP]
exact ⟨t.1 x, congr_fun t.w x⟩