-- 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
U₁ U₂ V : Opens (↑X : Type x)
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
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))
h₁ :
  Eq (α := CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op U₁)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map i₁.op) :
          (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op U₁)))
        s :
      CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op U₁)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map i₁.op) :
          (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op U₁)))
        t :
      CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op U₁)))
h₂ :
  Eq (α := CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op U₂)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map i₂.op) :
          (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op U₂)))
        s :
      CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op U₂)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map i₂.op) :
          (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op U₂)))
        t :
      CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj (op U₂)))
iUV : (i : Bool) → (if i = true then U₁ else U₂) ⟶ V
hcover₁ : V ≤ ⨆ (t : Bool), if t = true then U₁ else U₂
h :
  ∀ (i : Bool),
    Eq (α :=
      CC
        (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj
          (op (if i = true then U₁ else U₂))))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
                (Quiver.Hom.op (if h : i = true then eqToHom ⋯ ≫ i₁ else eqToHom ⋯ ≫ i₂))) :
            (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (if i = true then U₁ else U₂))))
          s :
        CC
          (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj
            (op (if i = true then U₁ else U₂))))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
                (Quiver.Hom.op (if h : i = true then eqToHom ⋯ ≫ i₁ else eqToHom ⋯ ≫ i₂))) :
            (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (if i = true then U₁ else U₂))))
          t :
        CC
          (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj
            (op (if i = true then U₁ else U₂))))
hcover₂ : V ≤ ⨆ (t : Bool), if t = true then U₁ else U₂
h₃ h₄ :
  ∀ (i : Bool),
    Eq (α :=
      CC
        (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj
          (op (if i = true then U₁ else U₂))))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
                (Quiver.Hom.op (if h : i = true then eqToHom ⋯ ≫ i₁ else eqToHom ⋯ ≫ i₂))) :
            (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (if i = true then U₁ else U₂))))
          s :
        CC
          (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj
            (op (if i = true then U₁ else U₂))))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).map
                (Quiver.Hom.op (if h : i = true then eqToHom ⋯ ≫ i₁ else eqToHom ⋯ ≫ i₂))) :
            (a : CC (F.obj.obj (op V))) → CC (F.obj.obj (op (if i = true then U₁ else U₂))))
          t :
        CC
          (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type x))).obj
            (op (if i = true then U₁ else U₂))))
⊢ s = t

Tactic:
classical
    fapply F.eq_of_locally_eq' fun t : Bool => if t then U₁ else U₂
    · exact fun i => if h : i then eqToHom (if_pos h) ≫ i₁ else eqToHom (if_neg h) ≫ i₂
    · refine le_trans hcover ?_
      rw [sup_le_iff]
      constructor
      · exact le_iSup (fun t : Bool => if t then U₁ else U₂) true
      · exact le_iSup (fun t : Bool => if t then U₁ else U₂) false
    · rintro ⟨_ | _⟩
      any_goals exact h₁
      any_goals exact h₂

Premises used: le_iSup, Bool.true, sup_le_iff, TopCat.Sheaf.eq_of_locally_eq', Bool.false, Eq, le_trans, ite