awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst_1 : (X Y : C) → FunLike.{u_2 + 1, u_3 + 1, u_3 + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory.{u_3, v_1, u_1, u_2} C FC (CC := CC)
inst_3 : HasLimitsOfSize.{x, x, v_1, u_1} C
inst_4 : Functor.ReflectsIsomorphisms.{v_1, u_1, u_3, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
inst_5 : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
X : TopCat
F : Sheaf.{x, v_1, u_1} C X
ι : Type u_4
U : ι → Opens (↑X : Type x)
V : Opens (↑X : Type x)
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
sf :
  (i : ι) →
    ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
      (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i)))
h :
  IsCompatible (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x)))) U sf
V_eq_supr_U : V = iSup U
gl :
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U)))
gl_spec :
  IsGluing (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x)))) U sf gl
gl_uniq :
  ∀
    (y :
      ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
        (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U)))),
    (fun
            (s :
              ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
                (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj
                  (op (iSup U)))) ↦
          IsGluing (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x)))) U sf s)
        y →
      y = gl
gl' :
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op V))
gl'_spec :
  ∀ (i : ι),
    (ConcreteCategory.hom (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map (iUV i).op) :
          (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (U i))))
        gl' =
      sf i
⊢ gl' =
    (ConcreteCategory.hom (FC := FC) (CC := CC)
          (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
            (Quiver.Hom.op (eqToHom ⋯) ≫ Quiver.Hom.op (eqToHom V_eq_supr_U))) :
        (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op V)))
      gl'

Tactic:
simp
-- Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst_1 : (X Y : C) → FunLike.{u_2 + 1, u_3 + 1, u_3 + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory.{u_3, v_1, u_1, u_2} C FC (CC := CC)
inst_3 : HasLimitsOfSize.{x, x, v_1, u_1} C
inst_4 : Functor.ReflectsIsomorphisms.{v_1, u_1, u_3, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
inst_5 : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
X : TopCat
F : Sheaf.{x, v_1, u_1} C X
ι : Type u_4
U : ι → Opens (↑X : Type x)
V : Opens (↑X : Type x)
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
sf :
  (i : ι) →
    ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
      (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i)))
h :
  IsCompatible (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x)))) U sf
V_eq_supr_U : V = iSup U
gl :
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U)))
gl_spec :
  IsGluing (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x)))) U sf gl
gl_uniq :
  ∀
    (y :
      ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
        (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U)))),
    (fun
            (s :
              ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
                (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj
                  (op (iSup U)))) ↦
          IsGluing (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x)))) U sf s)
        y →
      y = gl
gl' :
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op V))
gl'_spec :
  ∀ (i : ι),
    (ConcreteCategory.hom (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map (iUV i).op) :
          (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (U i))))
        gl' =
      sf i
i : ι
⊢ (ConcreteCategory.hom (FC := FC) (CC := CC)
          (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
            (Quiver.Hom.op (eqToHom ⋯) ≫ (leSupr U i).op)) :
        (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (U i))))
      gl' =
    sf i

Tactic:
gl'_spec i

Premises used: Set.Mem
-- Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst_1 : (X Y : C) → FunLike.{u_2 + 1, u_3 + 1, u_3 + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory.{u_3, v_1, u_1, u_2} C FC (CC := CC)
inst_3 : HasLimitsOfSize.{x, x, v_1, u_1} C
inst_4 : Functor.ReflectsIsomorphisms.{v_1, u_1, u_3, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
inst_5 : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
X : TopCat
F : Sheaf.{x, v_1, u_1} C X
ι : Type u_4
U : ι → Opens (↑X : Type x)
V : Opens (↑X : Type x)
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
sf :
  (i : ι) →
    ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
      (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i)))
h :
  IsCompatible (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x)))) U sf
V_eq_supr_U : V = iSup U
gl :
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U)))
gl_spec :
  IsGluing (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x)))) U sf gl
gl_uniq :
  ∀
    (y :
      ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
        (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U)))),
    (fun
            (s :
              ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
                (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj
                  (op (iSup U)))) ↦
          IsGluing (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x)))) U sf s)
        y →
      y = gl
gl' :
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op V))
gl'_spec :
  ∀ (i : ι),
    (ConcreteCategory.hom (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map (iUV i).op) :
          (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (U i))))
        gl' =
      sf i
i : ι
⊢ (ConcreteCategory.hom (FC := FC) (CC := CC)
          (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
            (Quiver.Hom.op (eqToHom ⋯) ≫ (leSupr U i).op)) :
        (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (U i))))
      gl' =
    sf i

Tactic:
exact gl'_spec i
-- Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst_1 : (X Y : C) → FunLike.{u_2 + 1, u_3 + 1, u_3 + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory.{u_3, v_1, u_1, u_2} C FC (CC := CC)
inst_3 : HasLimitsOfSize.{x, x, v_1, u_1} C
inst_4 : Functor.ReflectsIsomorphisms.{v_1, u_1, u_3, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
inst_5 : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
X : TopCat
F : Sheaf.{x, v_1, u_1} C X
ι : Type u_4
U : ι → Opens (↑X : Type x)
s t :
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U)))
h :
  ∀ (i : ι),
    Eq (α :=
      CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
                (leSupr U i).op) :
            (a : CC (F.obj.obj (op (iSup U)))) → CC (F.obj.obj (op (U i))))
          s :
        CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
                (leSupr U i).op) :
            (a : CC (F.obj.obj (op (iSup U)))) → CC (F.obj.obj (op (U i))))
          t :
        CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
sf : (i : ι) →
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))) :=
  fun (i : ι) ↦
    (ConcreteCategory.hom (FC := FC) (CC := CC)
          (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
            (leSupr U i).op) :
        (a : CC (F.obj.obj (op (iSup U)))) → CC (F.obj.obj (op (U i))))
      s
⊢ IsCompatible (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x)))) U sf

Tactic:
have sf_compatible : IsCompatible _ U sf := by
    intro i j
    simp_rw [sf, ← ConcreteCategory.comp_apply, ← F.1.map_comp]
    rfl
-- Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst_1 : (X Y : C) → FunLike.{u_2 + 1, u_3 + 1, u_3 + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory.{u_3, v_1, u_1, u_2} C FC (CC := CC)
inst_3 : HasLimitsOfSize.{x, x, v_1, u_1} C
inst_4 : Functor.ReflectsIsomorphisms.{v_1, u_1, u_3, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
inst_5 : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
X : TopCat
F : Sheaf.{x, v_1, u_1} C X
ι : Type u_4
U : ι → Opens (↑X : Type x)
s t :
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U)))
h :
  ∀ (i : ι),
    Eq (α :=
      CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
                (leSupr U i).op) :
            (a : CC (F.obj.obj (op (iSup U)))) → CC (F.obj.obj (op (U i))))
          s :
        CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
                (leSupr U i).op) :
            (a : CC (F.obj.obj (op (iSup U)))) → CC (F.obj.obj (op (U i))))
          t :
        CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
sf : (i : ι) →
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))) :=
  fun (i : ι) ↦
    (ConcreteCategory.hom (FC := FC) (CC := CC)
          (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
            (leSupr U i).op) :
        (a : CC (F.obj.obj (op (iSup U)))) → CC (F.obj.obj (op (U i))))
      s
sf_compatible :
  IsCompatible (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x)))) U sf
gl :
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U)))
gl_uniq :
  ∀
    (y :
      ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
        (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U)))),
    (fun
            (s :
              ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
                (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj
                  (op (iSup U)))) ↦
          IsGluing (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x)))) U sf s)
        y →
      y = gl
trans : s = gl
trans₁ : gl = t
⊢ s = t

Tactic:
trans gl
-- Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst_1 : (X Y : C) → FunLike.{u_2 + 1, u_3 + 1, u_3 + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory.{u_3, v_1, u_1, u_2} C FC (CC := CC)
inst_3 : HasLimitsOfSize.{x, x, v_1, u_1} C
inst_4 : Functor.ReflectsIsomorphisms.{v_1, u_1, u_3, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
inst_5 : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
X : TopCat
F : Sheaf.{x, v_1, u_1} C X
ι : Type u_4
U : ι → Opens (↑X : Type x)
s t :
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U)))
h :
  ∀ (i : ι),
    Eq (α :=
      CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
                (leSupr U i).op) :
            (a : CC (F.obj.obj (op (iSup U)))) → CC (F.obj.obj (op (U i))))
          s :
        CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
                (leSupr U i).op) :
            (a : CC (F.obj.obj (op (iSup U)))) → CC (F.obj.obj (op (U i))))
          t :
        CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
sf : (i : ι) →
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))) :=
  fun (i : ι) ↦
    (ConcreteCategory.hom (FC := FC) (CC := CC)
          (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
            (leSupr U i).op) :
        (a : CC (F.obj.obj (op (iSup U)))) → CC (F.obj.obj (op (U i))))
      s
sf_compatible :
  IsCompatible (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x)))) U sf
gl :
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U)))
gl_uniq :
  ∀
    (y :
      ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
        (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U)))),
    (fun
            (s :
              ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
                (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj
                  (op (iSup U)))) ↦
          IsGluing (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x)))) U sf s)
        y →
      y = gl
symm : t = gl
⊢ gl = t

Tactic:
symm
-- Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst_1 : (X Y : C) → FunLike.{u_2 + 1, u_3 + 1, u_3 + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory.{u_3, v_1, u_1, u_2} C FC (CC := CC)
inst_3 : HasLimitsOfSize.{x, x, v_1, u_1} C
inst_4 : Functor.ReflectsIsomorphisms.{v_1, u_1, u_3, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
inst_5 : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
X : TopCat
F : Sheaf.{x, v_1, u_1} C X
ι : Type u_4
U : ι → Opens (↑X : Type x)
V : Opens (↑X : Type x)
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
s t :
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op V))
h :
  ∀ (i : ι),
    Eq (α :=
      CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map (iUV i).op) :
            (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (U i))))
          s :
        CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map (iUV i).op) :
            (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (U i))))
          t :
        CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
⊢ V = iSup U

Tactic:
have V_eq_supr_U : V = iSup U := le_antisymm hcover (iSup_le fun i => (iUV i).le)
-- Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst_1 : (X Y : C) → FunLike.{u_2 + 1, u_3 + 1, u_3 + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory.{u_3, v_1, u_1, u_2} C FC (CC := CC)
inst_3 : HasLimitsOfSize.{x, x, v_1, u_1} C
inst_4 : Functor.ReflectsIsomorphisms.{v_1, u_1, u_3, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
inst_5 : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C (FC := FC) (CC := CC))
X : TopCat
F : Sheaf.{x, v_1, u_1} C X
ι : Type u_4
U : ι → Opens (↑X : Type x)
V : Opens (↑X : Type x)
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
s t :
  ToType.{u_3, v_1, u_1, u_2} (FC := FC) (CC := CC)
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op V))
h :
  ∀ (i : ι),
    Eq (α :=
      CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map (iUV i).op) :
            (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (U i))))
          s :
        CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map (iUV i).op) :
            (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (U i))))
          t :
        CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (U i))))
V_eq_supr_U : V = iSup U
suffices :
  Eq (α :=
    CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U))))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
              (Quiver.Hom.op (eqToHom ⋯))) :
          (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (iSup U))))
        s :
      CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U))))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
              (Quiver.Hom.op (eqToHom ⋯))) :
          (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (iSup U))))
        t :
      CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op (iSup U))))
⊢ s = t

Tactic:
suffices F.1.map (eqToHom V_eq_supr_U.symm).op s = F.1.map (eqToHom V_eq_supr_U.symm).op t by
    convert! congr_arg (F.1.map (eqToHom V_eq_supr_U).op) this <;>
    rw [← ConcreteCategory.comp_apply, ← F.1.map_comp, eqToHom_op, eqToHom_op, eqToHom_trans,
      eqToHom_refl, F.1.map_id, ConcreteCategory.id_apply]