awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
⊢ IsFiltered (CostructuredArrow yoneda (colimit F))
have refine : ∀ {J : Type v} {x : SmallCategory J} {x_1 : FinCategory J} (G : J ⥤ CostructuredArrow yoneda (colimit F)),
  ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
Tactic: refine IsFiltered.iff_nonempty_limit.mpr (fun {J _ _} G => ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x : SmallCategory J
x_1 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
⊢ ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
have h₁ : Nonempty
  (limit (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙ obj yoneda (Over.mk (𝟙 (colimit F)))))
Tactic: have h₁ : Nonempty (limit (G.op ⋙ (CostructuredArrow.toOver _ _).op ⋙
      yoneda.obj (Over.mk (𝟙 (colimit F))))) :=
    ⟨Types.Limit.mk _ (fun j => Over.mkIdTerminal.from _) (by simp)⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x : SmallCategory J
x_1 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
h₁ :
  Nonempty
    (limit (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙ obj yoneda (Over.mk (𝟙 (colimit F)))))
⊢ ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
have obtain : ∀ (i : I)
  (hi :
    Nonempty
      (limit
        ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙
          obj yoneda (obj (colimit.toCostructuredArrow F ⋙ CostructuredArrow.toOver F (colimit F)) i)))),
  ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
Tactic: obtain ⟨i, hi⟩ := exists_nonempty_limit_obj_of_isColimit F G _
    (colimit.isColimitToOver F) _ (Iso.refl _) h₁
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x : SmallCategory J
x_1 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
h₁ :
  Nonempty
    (limit (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙ obj yoneda (Over.mk (𝟙 (colimit F)))))
i : I
hi :
  Nonempty
    (limit
      ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙
        obj yoneda (obj (colimit.toCostructuredArrow F ⋙ CostructuredArrow.toOver F (colimit F)) i)))
⊢ ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
have obtain : ∀ (P : IndObjectPresentation (F.obj i)),
  ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
Tactic: obtain ⟨⟨P⟩⟩ := hF i
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x : SmallCategory J
x_1 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
h₁ :
  Nonempty
    (limit (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙ obj yoneda (Over.mk (𝟙 (colimit F)))))
i : I
hi :
  Nonempty
    (limit
      ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙
        obj yoneda (obj (colimit.toCostructuredArrow F ⋙ CostructuredArrow.toOver F (colimit F)) i)))
P : IndObjectPresentation (F.obj i)
hc : IsColimit (C := Over (colimit F)) (F :=
  (P.cocone.toCostructuredArrow ⋙ CostructuredArrow.toOver (P.F ⋙ yoneda) P.cocone.pt) ⋙ Over.map (colimit.ι F i))
  (mapCocone (Over.map (colimit.ι F i)) P.cocone.toOver) :=
  isColimitOfPreserves (Over.map (colimit.ι F i)) (Over.isColimitToOver P.coconeIsColimit)
⊢ ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
have obtain : ∀ (k : P.I)
  (hk :
    Nonempty
      (limit
        (G.op ⋙
          Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙
            obj yoneda
              (obj (CostructuredArrow.toOver yoneda (colimit F))
                (obj (CostructuredArrow.pre P.F yoneda (colimit F))
                  (obj (CostructuredArrow.map (colimit.ι F i))
                    (CostructuredArrow.mk (P.cocone.toCostructuredArrow.obj k).hom))))))),
  ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
Tactic: obtain ⟨k, hk⟩ : ∃ k, Nonempty (limit (G.op ⋙ (CostructuredArrow.toOver yoneda (colimit F)).op ⋙
      yoneda.obj ((CostructuredArrow.toOver yoneda (colimit F)).obj <|
        (CostructuredArrow.pre P.F yoneda (colimit F)).obj <|
          (map (colimit.ι F i)).obj <| mk _))) :=
    exists_nonempty_limit_obj_of_isColimit F G _ hc _ (Iso.refl _) hi
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x : SmallCategory J
x_1 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
h₁ :
  Nonempty
    (limit (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙ obj yoneda (Over.mk (𝟙 (colimit F)))))
i : I
hi :
  Nonempty
    (limit
      ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙
        obj yoneda (obj (colimit.toCostructuredArrow F ⋙ CostructuredArrow.toOver F (colimit F)) i)))
P : IndObjectPresentation (F.obj i)
hc : IsColimit (C := Over (colimit F)) (F :=
  (P.cocone.toCostructuredArrow ⋙ CostructuredArrow.toOver (P.F ⋙ yoneda) P.cocone.pt) ⋙ Over.map (colimit.ι F i))
  (mapCocone (Over.map (colimit.ι F i)) P.cocone.toOver) :=
  isColimitOfPreserves (Over.map (colimit.ι F i)) (Over.isColimitToOver P.coconeIsColimit)
k : P.I
hk :
  Nonempty
    (limit
      (G.op ⋙
        Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙
          obj yoneda
            (obj (CostructuredArrow.toOver yoneda (colimit F))
              (obj (CostructuredArrow.pre P.F yoneda (colimit F))
                (obj (CostructuredArrow.map (colimit.ι F i))
                  (CostructuredArrow.mk (P.cocone.toCostructuredArrow.obj k).hom))))))
⊢ ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
have have : ∀ (htO : FullyFaithful (C := CostructuredArrow yoneda (colimit F)) (CostructuredArrow.toOver yoneda (colimit F))),
  ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
Tactic: have htO : (CostructuredArrow.toOver yoneda (colimit F)).FullyFaithful := .ofFullyFaithful _
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
inst_2 : IsFiltered I
hF : ∀ (i : I), IsIndObject (F.obj i)
J : Type v
x : SmallCategory J
x_1 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
h₁ :
  Nonempty
    (limit (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙ obj yoneda (Over.mk (𝟙 (colimit F)))))
i : I
hi :
  Nonempty
    (limit
      ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙
        obj yoneda (obj (colimit.toCostructuredArrow F ⋙ CostructuredArrow.toOver F (colimit F)) i)))
P : IndObjectPresentation (F.obj i)
hc : IsColimit (C := Over (colimit F)) (F :=
  (P.cocone.toCostructuredArrow ⋙ CostructuredArrow.toOver (P.F ⋙ yoneda) P.cocone.pt) ⋙ Over.map (colimit.ι F i))
  (mapCocone (Over.map (colimit.ι F i)) P.cocone.toOver) :=
  isColimitOfPreserves (Over.map (colimit.ι F i)) (Over.isColimitToOver P.coconeIsColimit)
k : P.I
hk :
  Nonempty
    (limit
      (G.op ⋙
        Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙
          obj yoneda
            (obj (CostructuredArrow.toOver yoneda (colimit F))
              (obj (CostructuredArrow.pre P.F yoneda (colimit F))
                (obj (CostructuredArrow.map (colimit.ι F i))
                  (CostructuredArrow.mk (P.cocone.toCostructuredArrow.obj k).hom))))))
htO : FullyFaithful (C := CostructuredArrow yoneda (colimit F)) (CostructuredArrow.toOver yoneda (colimit F))
q : (X : CostructuredArrow yoneda (colimit F)) →
  Functor.op (CostructuredArrow.toOver yoneda (colimit F)) ⋙
      obj yoneda (obj (CostructuredArrow.toOver yoneda (colimit F)) X) ≅
    obj uliftYoneda.{max u v, v, max u v} X :=
  fun (X : CostructuredArrow yoneda (colimit F)) ↦
    isoWhiskerLeft (Functor.op (CostructuredArrow.toOver yoneda (colimit F)))
        (Iso.app (Iso.symm uliftYonedaIsoYoneda) (obj (CostructuredArrow.toOver yoneda (colimit F)) X)) ≪≫
      htO.homNatIso X
⊢ ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
have obtain : ∀
  (t' :
    limit
      (G.op ⋙
        obj uliftYoneda.{max u v, v, max u v}
          (obj (CostructuredArrow.pre P.F yoneda (colimit F))
            (obj (CostructuredArrow.map (colimit.ι F i))
              (CostructuredArrow.mk (P.cocone.toCostructuredArrow.obj k).hom))))),
  ∃ (X : CostructuredArrow yoneda (colimit F)), Nonempty.{v + 1} (limit (G.op ⋙ obj yoneda X))
Tactic: obtain ⟨t'⟩ := Nonempty.map (limMap (isoWhiskerLeft G.op (q _)).hom) hk
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
hF : ∀ (i : I), IsIndObject (F.obj i)
⊢ IsIndObject (colimit F)
have this : IsFiltered (CostructuredArrow yoneda (colimit F))
Tactic: have : IsFiltered (CostructuredArrow yoneda (colimit F)) :=
    IndizationClosedUnderFilteredColimitsAux.isFiltered F hF
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
⊢ IsIndObject (colimit F)
have refine : FinallySmall (CostructuredArrow yoneda (colimit F))
Tactic: refine (isIndObject_iff _).mpr ⟨this, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
⊢ FinallySmall (CostructuredArrow yoneda (colimit F))
have this₁ : ∀ (i : I),
  ∃ (s : Set (CostructuredArrow yoneda (F.obj i))) (_ : Small.{v, max u v} (↑s : Type (max u v))),
    ∀ (i_1 : CostructuredArrow yoneda (F.obj i)), ∃ j ∈ s, Nonempty.{v + 1} (i_1 ⟶ j)
Tactic: have : ∀ i, ∃ (s : Set (CostructuredArrow yoneda (F.obj i))) (_ : Small.{v} s),
      ∀ i, ∃ j ∈ s, Nonempty (i ⟶ j) :=
    fun i => (hF i).finallySmall.exists_small_weakly_terminal_set
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
hF : ∀ (i : I), IsIndObject (F.obj i)
this_1 : IsFiltered (CostructuredArrow yoneda (colimit F))
this :
  ∀ (i : I),
    ∃ (s : Set (CostructuredArrow yoneda (F.obj i))) (_ : Small.{v, max u v} (↑s : Type (max u v))),
      ∀ (i_1 : CostructuredArrow yoneda (F.obj i)), ∃ j ∈ s, Nonempty.{v + 1} (i_1 ⟶ j)
⊢ FinallySmall (CostructuredArrow yoneda (colimit F))
have choose : ∀ (s : (i : I) → Set (CostructuredArrow yoneda (F.obj i)))
  (hs : ∀ (i : I), Small.{v, max u v} (↑(s i) : Type (max u v)))
  (j : (i : I) → (i_1 : CostructuredArrow yoneda (F.obj i)) → CostructuredArrow yoneda (F.obj i))
  (hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i)
  (hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty.{v + 1} (i_1 ⟶ j i i_1)),
  FinallySmall (CostructuredArrow yoneda (colimit F))
Tactic: choose s hs j hjs hj using this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} (↑(s i) : Type (max u v))
j : (i : I) → (i_1 : CostructuredArrow yoneda (F.obj i)) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty.{v + 1} (i_1 ⟶ j i i_1)
⊢ FinallySmall (CostructuredArrow yoneda (colimit F))
have refine : ∀ (A : CostructuredArrow yoneda (colimit F)),
  ∃ (j : CostructuredArrow yoneda (colimit F)),
    Membership.mem (γ := Set (CostructuredArrow yoneda (colimit F)))
        (⋃ (i : I), obj (CostructuredArrow.map (colimit.ι F i)) '' s i) j ∧
      Nonempty.{v + 1} (A ⟶ j)
Tactic: refine finallySmall_of_small_weakly_terminal_set
    (⋃ i, (map (colimit.ι F i)).obj '' (s i)) (fun A => ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} (↑(s i) : Type (max u v))
j : (i : I) → (i_1 : CostructuredArrow yoneda (F.obj i)) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty.{v + 1} (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
⊢ ∃ (j : CostructuredArrow yoneda (colimit F)),
    Membership.mem (γ := Set (CostructuredArrow yoneda (colimit F)))
        (⋃ (i : I), obj (CostructuredArrow.map (colimit.ι F i)) '' s i) j ∧
      Nonempty.{v + 1} (A ⟶ j)
have obtain : ∀ (i : I) (y : (F.obj i).obj (Opposite.op A.left))
  (hy :
    Eq.{v + 1} (α := (colimit F).obj (Opposite.op A.left))
      ((yonedaEquiv : (a : yoneda.obj A.left ⟶ colimit F) → (colimit F).obj (Opposite.op A.left)) A.hom :
        (colimit F).obj (Opposite.op A.left))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (NatTrans.app (colimit.ι F i) (Opposite.op A.left)) :
            (a : (fun (X : Type v) ↦ X) ((F.obj i).obj (Opposite.op A.left))) →
              (fun (X : Type v) ↦ X) ((colimit F).obj (Opposite.op A.left)))
          y :
        (fun (X : Type v) ↦ X) ((colimit F).obj (Opposite.op A.left)))),
  ∃ (j : CostructuredArrow yoneda (colimit F)),
    Membership.mem (γ := Set (CostructuredArrow yoneda (colimit F)))
        (⋃ (i : I), obj (CostructuredArrow.map (colimit.ι F i)) '' s i) j ∧
      Nonempty.{v + 1} (A ⟶ j)
Tactic: obtain ⟨i, y, hy⟩ := FunctorToTypes.jointly_surjective'.{v, v} F _ (yonedaEquiv A.hom)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} (↑(s i) : Type (max u v))
j : (i : I) → (i_1 : CostructuredArrow yoneda (F.obj i)) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty.{v + 1} (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
i : I
y : (F.obj i).obj (Opposite.op A.left)
hy :
  Eq.{v + 1} (α := (colimit F).obj (Opposite.op A.left))
    ((yonedaEquiv : (a : yoneda.obj A.left ⟶ colimit F) → (colimit F).obj (Opposite.op A.left)) A.hom :
      (colimit F).obj (Opposite.op A.left))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (NatTrans.app (colimit.ι F i) (Opposite.op A.left)) :
          (a : (fun (X : Type v) ↦ X) ((F.obj i).obj (Opposite.op A.left))) →
            (fun (X : Type v) ↦ X) ((colimit F).obj (Opposite.op A.left)))
        y :
      (fun (X : Type v) ↦ X) ((colimit F).obj (Opposite.op A.left)))
y' : CostructuredArrow yoneda (F.obj i) :=
  CostructuredArrow.mk (Y := A.left)
    ((Equiv.symm yonedaEquiv : (a : (F.obj i).obj (Opposite.op A.left)) → yoneda.obj A.left ⟶ F.obj i) y)
⊢ ∃ (j : CostructuredArrow yoneda (colimit F)),
    Membership.mem (γ := Set (CostructuredArrow yoneda (colimit F)))
        (⋃ (i : I), obj (CostructuredArrow.map (colimit.ι F i)) '' s i) j ∧
      Nonempty.{v + 1} (A ⟶ j)
have obtain : ∀ (x : y' ⟶ j i y'),
  ∃ (j : CostructuredArrow yoneda (colimit F)),
    Membership.mem (γ := Set (CostructuredArrow yoneda (colimit F)))
        (⋃ (i : I), obj (CostructuredArrow.map (colimit.ι F i)) '' s i) j ∧
      Nonempty.{v + 1} (A ⟶ j)
Tactic: obtain ⟨x⟩ := hj _ y'
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} (↑(s i) : Type (max u v))
j : (i : I) → (i_1 : CostructuredArrow yoneda (F.obj i)) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty.{v + 1} (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
i : I
y : (F.obj i).obj (Opposite.op A.left)
hy :
  Eq.{v + 1} (α := (colimit F).obj (Opposite.op A.left))
    ((yonedaEquiv : (a : yoneda.obj A.left ⟶ colimit F) → (colimit F).obj (Opposite.op A.left)) A.hom :
      (colimit F).obj (Opposite.op A.left))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (NatTrans.app (colimit.ι F i) (Opposite.op A.left)) :
          (a : (fun (X : Type v) ↦ X) ((F.obj i).obj (Opposite.op A.left))) →
            (fun (X : Type v) ↦ X) ((colimit F).obj (Opposite.op A.left)))
        y :
      (fun (X : Type v) ↦ X) ((colimit F).obj (Opposite.op A.left)))
y' : CostructuredArrow yoneda (F.obj i) :=
  CostructuredArrow.mk (Y := A.left)
    ((Equiv.symm yonedaEquiv : (a : (F.obj i).obj (Opposite.op A.left)) → yoneda.obj A.left ⟶ F.obj i) y)
x : y' ⟶ j i y'
⊢ ∃ (j : CostructuredArrow yoneda (colimit F)),
    Membership.mem (γ := Set (CostructuredArrow yoneda (colimit F)))
        (⋃ (i : I), obj (CostructuredArrow.map (colimit.ι F i)) '' s i) j ∧
      Nonempty.{v + 1} (A ⟶ j)
have refine_1 : Membership.mem (γ := Set (CostructuredArrow yoneda (colimit F)))
  (⋃ (i : I), obj (CostructuredArrow.map (colimit.ι F i)) '' s i) (obj (CostructuredArrow.map (colimit.ι F i)) (j i y'))
Tactic: refine ⟨(map (colimit.ι F i)).obj (j i y'), ?_, ⟨?_⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} (↑(s i) : Type (max u v))
j : (i : I) → (i_1 : CostructuredArrow yoneda (F.obj i)) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty.{v + 1} (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
i : I
y : (F.obj i).obj (Opposite.op A.left)
hy :
  Eq.{v + 1} (α := (colimit F).obj (Opposite.op A.left))
    ((yonedaEquiv : (a : yoneda.obj A.left ⟶ colimit F) → (colimit F).obj (Opposite.op A.left)) A.hom :
      (colimit F).obj (Opposite.op A.left))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (NatTrans.app (colimit.ι F i) (Opposite.op A.left)) :
          (a : (fun (X : Type v) ↦ X) ((F.obj i).obj (Opposite.op A.left))) →
            (fun (X : Type v) ↦ X) ((colimit F).obj (Opposite.op A.left)))
        y :
      (fun (X : Type v) ↦ X) ((colimit F).obj (Opposite.op A.left)))
y' : CostructuredArrow yoneda (F.obj i) :=
  CostructuredArrow.mk (Y := A.left)
    ((Equiv.symm yonedaEquiv : (a : (F.obj i).obj (Opposite.op A.left)) → yoneda.obj A.left ⟶ F.obj i) y)
x : y' ⟶ j i y'
refine_1 :
  Membership.mem (γ := Set (CostructuredArrow yoneda (colimit F)))
    (⋃ (i : I), obj (CostructuredArrow.map (colimit.ι F i)) '' s i)
    (obj (CostructuredArrow.map (colimit.ι F i)) (j i y'))
⊢ ∃ (j : CostructuredArrow yoneda (colimit F)),
    Membership.mem (γ := Set (CostructuredArrow yoneda (colimit F)))
        (⋃ (i : I), obj (CostructuredArrow.map (colimit.ι F i)) '' s i) j ∧
      Nonempty.{v + 1} (A ⟶ j)
have refine_2 : A ⟶ obj (CostructuredArrow.map (colimit.ι F i)) (j i y')
Tactic: refine ⟨(map (colimit.ι F i)).obj (j i y'), ?_, ⟨?_⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Indization.FilteredColimits (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
inst_2 : IsFiltered I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} (↑(s i) : Type (max u v))
j : (i : I) → (i_1 : CostructuredArrow yoneda (F.obj i)) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty.{v + 1} (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
i : I
y : (F.obj i).obj (Opposite.op A.left)
hy :
  Eq.{v + 1} (α := (colimit F).obj (Opposite.op A.left))
    ((yonedaEquiv : (a : yoneda.obj A.left ⟶ colimit F) → (colimit F).obj (Opposite.op A.left)) A.hom :
      (colimit F).obj (Opposite.op A.left))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (NatTrans.app (colimit.ι F i) (Opposite.op A.left)) :
          (a : (fun (X : Type v) ↦ X) ((F.obj i).obj (Opposite.op A.left))) →
            (fun (X : Type v) ↦ X) ((colimit F).obj (Opposite.op A.left)))
        y :
      (fun (X : Type v) ↦ X) ((colimit F).obj (Opposite.op A.left)))
y' : CostructuredArrow yoneda (F.obj i) :=
  CostructuredArrow.mk (Y := A.left)
    ((Equiv.symm yonedaEquiv : (a : (F.obj i).obj (Opposite.op A.left)) → yoneda.obj A.left ⟶ F.obj i) y)
x : y' ⟶ j i y'
⊢ Membership.mem (γ := Set (CostructuredArrow yoneda (colimit F)))
    (⋃ (i : I), obj (CostructuredArrow.map (colimit.ι F i)) '' s i)
    (obj (CostructuredArrow.map (colimit.ι F i)) (j i y'))
have refine_1 : ∃ (i_1 : I),
  ∃ x ∈ s i_1,
    Eq (α := CostructuredArrow yoneda (colimit F)) (obj (CostructuredArrow.map (colimit.ι F i_1)) x)
      (obj (CostructuredArrow.map (colimit.ι F i)) (j i y'))
Tactic: simp only [Set.mem_iUnion, Set.mem_image]