awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
i j k : D.J
U : Opens (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)
x : (↑(↑(D.V (k, i)) : TopCat) : Type v)
y : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)
eq :
  (ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) (X :=
          (↑(pullback (D.f i j) (D.f i k)) : TopCat)) (Hom.base (pullback.snd (D.f i j) (D.f i k) ≫ D.t i k)) :
        (a : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)) → (↑(↑(D.V (k, i)) : TopCat) : Type v))
      y =
    x
h :
  Membership.mem (α := (↑(↑(D.V (k, i)) : TopCat) : Type v)) (γ := Set (↑(↑(D.V (k, i)) : TopCat) : Type v))
    (Set.range (ι := (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (↑(pullback (D.f k i) (D.f k j)) : TopCat)) (Hom.base (pullback.fst (D.f k i) (D.f k j)))) :
        (a : (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v)) → (↑(↑(D.V (k, i)) : TopCat) : Type v)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) (X :=
            (↑(pullback (D.f i j) (D.f i k)) : TopCat)) (Hom.base (pullback.snd (D.f i j) (D.f i k) ≫ D.t i k)) :
          (a : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)) → (↑(↑(D.V (k, i)) : TopCat) : Type v))
        y :
      (↑(↑(D.V (k, i)) : TopCat) : Type v))
⊢ Membership.mem (γ := Set (↑(↑(D.V (k, i)) : TopCat) : Type v))
    (Set.range (ι := (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (↑(pullback (D.f k i) (D.f k j)) : TopCat)) (Hom.base (pullback.fst (D.f k i) (D.f k j)))) :
        (a : (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v)) → (↑(↑(D.V (k, i)) : TopCat) : Type v)))
    x

Tactic:
subst eq
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
i j k : D.J
U : Opens (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)
y : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)
h :
  Eq (α := (↑(↑(D.V (k, i)) : TopCat) : Type v))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) (X :=
            (↑(pullback (D.f k i) (D.f k j)) : TopCat)) (Hom.base (pullback.fst (D.f k i) (D.f k j))) :
          (a : (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v)) → (↑(↑(D.V (k, i)) : TopCat) : Type v))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (inv (D.t' k i j)).base :
              (a : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)) →
                (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v))
            y :
          (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v)) :
      (↑(↑(D.V (k, i)) : TopCat) : Type v))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) (X :=
            (↑(pullback (D.f i j) (D.f i k)) : TopCat)) (Hom.base (pullback.snd (D.f i j) (D.f i k) ≫ D.t i k)) :
          (a : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)) → (↑(↑(D.V (k, i)) : TopCat) : Type v))
        y :
      (↑(↑(D.V (k, i)) : TopCat) : Type v))
⊢ Membership.mem (α := (↑(↑(D.V (k, i)) : TopCat) : Type v)) (γ := Set (↑(↑(D.V (k, i)) : TopCat) : Type v))
    (Set.range (ι := (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (↑(pullback (D.f k i) (D.f k j)) : TopCat)) (Hom.base (pullback.fst (D.f k i) (D.f k j)))) :
        (a : (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v)) → (↑(↑(D.V (k, i)) : TopCat) : Type v)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) (X :=
            (↑(pullback (D.f i j) (D.f i k)) : TopCat)) (Hom.base (pullback.snd (D.f i j) (D.f i k) ≫ D.t i k)) :
          (a : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)) → (↑(↑(D.V (k, i)) : TopCat) : Type v))
        y :
      (↑(↑(D.V (k, i)) : TopCat) : Type v))

Tactic:
use (inv (D.t' k i j)).base y

Premises used: CategoryTheory.ConcreteCategory.hom, CategoryTheory.GlueData.t', AlgebraicGeometry.PresheafedSpace.Hom.base, CategoryTheory.inv, DFunLike.coe, AlgebraicGeometry.PresheafedSpace.GlueData.toGlueData
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
i j k : D.J
U : Opens (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)
y : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)
h : inv (D.t' k i j) ≫ pullback.fst (D.f k i) (D.f k j) = pullback.snd (D.f i j) (D.f i k) ≫ D.t i k
⊢ Eq (α := (↑(↑(D.V (k, i)) : TopCat) : Type v))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (inv (D.t' k i j) ≫ pullback.fst (D.f k i) (D.f k j)).base :
          (a : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)) → (↑(↑(D.V (k, i)) : TopCat) : Type v))
        y :
      (↑(↑(D.V (k, i)) : TopCat) : Type v))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) (X :=
            (↑(pullback (D.f i j) (D.f i k)) : TopCat)) (Hom.base (pullback.snd (D.f i j) (D.f i k) ≫ D.t i k)) :
          (a : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)) → (↑(↑(D.V (k, i)) : TopCat) : Type v))
        y :
      (↑(↑(D.V (k, i)) : TopCat) : Type v))

Tactic:
congr 3
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
i j k : D.J
U : Opens (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)
y : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)
h : pullback.fst (D.f k i) (D.f k j) = D.t' k i j ≫ pullback.snd (D.f i j) (D.f i k) ≫ D.t i k
h₁ :
  Eq (α := pullback (D.f k i) (D.f k j) ⟶ D.V (k, i)) (pullback.fst (D.f k i) (D.f k j))
    (pullback.fst (D.f k i) (D.f k j) ≫ D.t k i ≫ D.t i k)
h₂ :
  Eq (α := pullback (D.f k i) (D.f k j) ⟶ D.V (k, i)) (pullback.fst (D.f k i) (D.f k j))
    (pullback.fst (D.f k i) (D.f k j) ≫ 𝟙 (D.V (k, i)))
h₃ h₄ :
  Eq (α := pullback (D.f k i) (D.f k j) ⟶ D.V (k, i)) (pullback.fst (D.f k i) (D.f k j))
    (pullback.fst (D.f k i) (D.f k j))
⊢ inv (D.t' k i j) ≫ pullback.fst (D.f k i) (D.f k j) = pullback.snd (D.f i j) (D.f i k) ≫ D.t i k

Tactic:
rw [IsIso.inv_comp_eq, 𝖣.t_fac_assoc, 𝖣.t_inv, Category.comp_id]

Premises used: CategoryTheory.IsIso.inv_comp_eq, CategoryTheory.Category.comp_id
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
i j k : D.J
U : Opens (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)
⊢ invApp (pullback.snd (D.f i j) (D.f i k)) U ≫
      (D.t k i).c.app (op (Functor.obj (opensFunctor (pullback.snd (D.f i j) (D.f i k))) U)) ≫
        (D.V (k, i)).presheaf.map (eqToHom ⋯) =
    (D.t' k i j).c.app (op U) ≫
      invApp (pullback.fst (D.f k i) (D.f k j)) (unop (op ((Opens.map (D.t' k i j).base).1 (unop (op U)))))

Tactic:
have e := (D.snd_invApp_t_app' i j k U).choose_spec
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
i j k : D.J
U : Opens (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)
e :
  invApp (pullback.snd (D.f i j) (D.f i k)) U ≫
      (D.t k i).c.app (op (Functor.obj (opensFunctor (pullback.snd (D.f i j) (D.f i k))) U)) ≫
        (D.V (k, i)).presheaf.map (eqToHom ⋯) =
    (D.t' k i j).c.app (op U) ≫
      invApp (pullback.fst (D.f k i) (D.f k j)) (unop (op ((Opens.map (D.t' k i j).base).1 (unop (op U)))))
⊢ ∀ {Z : C}
    (h :
      (D.V (k, i)).presheaf.obj
          (op
            (Functor.obj (C := Opens (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v))
              (opensFunctor (pullback.fst (D.f k i) (D.f k j))) ((Opens.map (D.t' k i j).base).1 U))) ⟶
        Z),
    invApp (pullback.snd (D.f i j) (D.f i k)) U ≫
        (D.t k i).c.app (op (Functor.obj (opensFunctor (pullback.snd (D.f i j) (D.f i k))) U)) ≫
          (D.V (k, i)).presheaf.map (eqToHom ⋯) ≫ h =
      (D.t' k i j).c.app (op U) ≫ invApp (pullback.fst (D.f k i) (D.f k j)) ((Opens.map (D.t' k i j).base).1 U) ≫ h

Tactic:
replace e := reassoc_of% e
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
i j k : D.J
U : Opens (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)
e :
  ∀ {Z : C}
    (h :
      (D.V (k, i)).presheaf.obj
          (op
            (Functor.obj (C := Opens (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v))
              (opensFunctor (pullback.fst (D.f k i) (D.f k j))) ((Opens.map (D.t' k i j).base).1 U))) ⟶
        Z),
    invApp (pullback.snd (D.f i j) (D.f i k)) U ≫
        (D.t k i).c.app (op (Functor.obj (opensFunctor (pullback.snd (D.f i j) (D.f i k))) U)) ≫
          (D.V (k, i)).presheaf.map (eqToHom ⋯) ≫ h =
      (D.t' k i j).c.app (op U) ≫ invApp (pullback.fst (D.f k i) (D.f k j)) ((Opens.map (D.t' k i j).base).1 U) ≫ h
rw :
  invApp (pullback.snd (D.f i j) (D.f i k)) U ≫
      (D.t k i).c.app (op (Functor.obj (opensFunctor (pullback.snd (D.f i j) (D.f i k))) U)) =
    invApp (pullback.snd (D.f i j) (D.f i k)) U ≫
      (D.t k i).c.app (op (Functor.obj (opensFunctor (pullback.snd (D.f i j) (D.f i k))) U)) ≫
        (D.V (k, i)).presheaf.map (eqToHom ⋯) ≫ (D.V (k, i)).presheaf.map (eqToHom ⋯)
⊢ invApp (pullback.snd (D.f i j) (D.f i k)) U ≫
      (D.t k i).c.app (op (Functor.obj (opensFunctor (pullback.snd (D.f i j) (D.f i k))) U)) =
    (D.t' k i j).c.app (op U) ≫
      invApp (pullback.fst (D.f k i) (D.f k j)) (unop (op ((Opens.map (D.t' k i j).base).1 (unop (op U))))) ≫
        (D.V (k, i)).presheaf.map (eqToHom ⋯)

Tactic:
rw [← e]
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
i j k : D.J
U : Opens (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)
e :
  ∀ {Z : C}
    (h :
      (D.V (k, i)).presheaf.obj
          (op
            (Functor.obj (C := Opens (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v))
              (opensFunctor (pullback.fst (D.f k i) (D.f k j))) ((Opens.map (D.t' k i j).base).1 U))) ⟶
        Z),
    invApp (pullback.snd (D.f i j) (D.f i k)) U ≫
        (D.t k i).c.app (op (Functor.obj (opensFunctor (pullback.snd (D.f i j) (D.f i k))) U)) ≫
          (D.V (k, i)).presheaf.map (eqToHom ⋯) ≫ h =
      (D.t' k i j).c.app (op U) ≫ invApp (pullback.fst (D.f k i) (D.f k j)) ((Opens.map (D.t' k i j).base).1 U) ≫ h
⊢ invApp (pullback.snd (D.f i j) (D.f i k)) U ≫
      (D.t k i).c.app (op (Functor.obj (opensFunctor (pullback.snd (D.f i j) (D.f i k))) U)) =
    invApp (pullback.snd (D.f i j) (D.f i k)) U ≫
      (D.t k i).c.app (op (Functor.obj (opensFunctor (pullback.snd (D.f i j) (D.f i k))) U)) ≫
        (D.V (k, i)).presheaf.map (eqToHom ⋯) ≫ (D.V (k, i)).presheaf.map (eqToHom ⋯)

Tactic:
simp [eqToHom_map]

Premises used: CategoryTheory.eqToHom_map
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
h :
  (↑(Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
          (Opens.map (Hom.base (D.ι j)) : Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
          (Functor.obj
              (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
              U :
            Opens (↑(↑D.glued : TopCat) : Type v))) :
      Set (↑(↑(D.U j) : TopCat) : Type v)) =
    (↑((opensFunctor (D.f j i)).obj ((Opens.map (D.t j i).base).obj ((Opens.map (D.f i j).base).obj U))) :
      Set (↑(↑(D.U j) : TopCat) : Type v))
⊢ Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
      (Opens.map (Hom.base (D.ι j)) : Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
      (Functor.obj
          (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
            Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
          U :
        Opens (↑(↑D.glued : TopCat) : Type v)) =
    (opensFunctor (D.f j i)).obj ((Opens.map (D.t j i).base).obj ((Opens.map (D.f i j).base).obj U))

Tactic:
ext1
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
h :
  Eq (α := Set (↑(↑(D.U j) : TopCat) : Type v))
    (Set.preimage (β := (↑(↑D.glued : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (Y := (↑D.glued : TopCat)) (Hom.base (D.ι j))) :
        (a : (↑(↑(D.U j) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (Y := (↑D.glued : TopCat)) (Hom.base (D.ι i))) :
          (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v)) ''
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
    (Set.image (α := (↑(↑(D.V (j, i)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.f j i).base) :
        (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.U j) : TopCat) : Type v))
      (Set.preimage (β := (↑(↑(D.V (i, j)) : TopCat) : Type v))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.t j i).base) :
          (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
        ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (D.f i j).base) :
            (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
          (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))))
⊢ (↑(Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
          (Opens.map (Hom.base (D.ι j)) : Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
          (Functor.obj
              (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
              U :
            Opens (↑(↑D.glued : TopCat) : Type v))) :
      Set (↑(↑(D.U j) : TopCat) : Type v)) =
    (↑((opensFunctor (D.f j i)).obj ((Opens.map (D.t j i).base).obj ((Opens.map (D.f i j).base).obj U))) :
      Set (↑(↑(D.U j) : TopCat) : Type v))

Tactic:
dsimp only [Opens.map_coe, IsOpenMap.coe_functor_obj]

Premises used: TopologicalSpace.Opens.map_coe, IsOpenMap.coe_functor_obj
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ Eq (α := Set (↑(↑(D.U j) : TopCat) : Type v))
    (Set.preimage (β := (↑(D.mapGlueData (forget C)).glued : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (Y := (D.mapGlueData (forget C)).glued) ((D.mapGlueData (forget C)).ι j)) :
        (a : (↑(↑(D.U j) : TopCat) : Type v)) → (↑(D.mapGlueData (forget C)).glued : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (Y := (D.mapGlueData (forget C)).glued) ((D.mapGlueData (forget C)).ι i)) :
          (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(D.mapGlueData (forget C)).glued : Type v)) ''
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
    (Set.image (α := (↑(↑(D.V (j, i)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.f j i).base) :
        (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.U j) : TopCat) : Type v))
      (Set.preimage (β := (↑(↑(D.V (i, j)) : TopCat) : Type v))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.t j i).base) :
          (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
        ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (D.f i j).base) :
            (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
          (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))))

Tactic:
Eq.trans (D.toTopGlueData.preimage_image_eq_image' _ _ _) ?_

Premises used: AlgebraicGeometry.PresheafedSpace.GlueData.toTopGlueData, Eq.trans, TopCat.GlueData.preimage_image_eq_image'
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ Eq (α := Set (↑(↑(D.U j) : TopCat) : Type v))
    (Set.image (α := (↑(↑(D.V (j, i)) : TopCat) : Type v))
      (⇑(TopCat.Hom.hom (D.f j i).base) : (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.U j) : TopCat) : Type v))
      (Set.image (α := (↑(↑(D.V (i, j)) : TopCat) : Type v))
        (⇑(TopCat.Hom.hom (D.t i j).base) :
          (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.V (j, i)) : TopCat) : Type v))
        ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (D.f i j).base) :
            (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
          (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))))
    (Set.image (α := (↑(↑(D.V (j, i)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.f j i).base) :
        (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.U j) : TopCat) : Type v))
      (Set.preimage (β := (↑(↑(D.V (i, j)) : TopCat) : Type v))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.t j i).base) :
          (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
        ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (D.f i j).base) :
            (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
          (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))))

Tactic:
congr_arg (_ '' ·) ?_

Premises used: congr_arg, Set.image
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
h :
  Eq (α := Set (↑(↑(D.V (j, i)) : TopCat) : Type v))
    (Set.image (α := (↑(↑(D.V (i, j)) : TopCat) : Type v))
      (⇑(TopCat.Hom.hom (D.t i j).base) :
        (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.V (j, i)) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.f i j).base) :
          (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
    (Set.preimage (β := (↑(↑(D.V (i, j)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.t j i).base) :
        (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.f i j).base) :
          (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
⊢ Eq (α := Set (↑(↑(D.U j) : TopCat) : Type v))
    (Set.image (α := (↑(↑(D.V (j, i)) : TopCat) : Type v))
      (⇑(TopCat.Hom.hom (D.f j i).base) : (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.U j) : TopCat) : Type v))
      (Set.image (α := (↑(↑(D.V (i, j)) : TopCat) : Type v))
        (⇑(TopCat.Hom.hom (D.t i j).base) :
          (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.V (j, i)) : TopCat) : Type v))
        ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (D.f i j).base) :
            (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
          (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))))
    (Set.image (α := (↑(↑(D.V (j, i)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.f j i).base) :
        (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.U j) : TopCat) : Type v))
      (Set.preimage (β := (↑(↑(D.V (i, j)) : TopCat) : Type v))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.t j i).base) :
          (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
        ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (D.f i j).base) :
            (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
          (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))))

Tactic:
refine congr_arg (_ '' ·) ?_

Premises used: congr_arg, Set.image