-- Module: Mathlib.AlgebraicGeometry.Sites.Representability
Goal:
F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → Functor.obj yoneda (X i) ⟶ F.obj (P := Presheaf.IsSheaf zariskiTopology)
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i j k : ι
U : Scheme
α β : U ⟶ (glueData hf).glued
h :
  Eq (α := (fun (X : Type u) ↦ X) (F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op U)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((yonedaGluedToSheaf hf).hom.app (op U)) :
          (a : (fun (X : Type u) ↦ X) ((zariskiTopology.yoneda.obj (glueData hf).glued).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) (F.obj.obj (op U)))
        α :
      (fun (X : Type u) ↦ X) (F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op U)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((yonedaGluedToSheaf hf).hom.app (op U)) :
          (a : (fun (X : Type u) ↦ X) ((zariskiTopology.yoneda.obj (glueData hf).glued).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) (F.obj.obj (op U)))
        β :
      (fun (X : Type u) ↦ X) (F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op U)))
mem :
  Membership.mem (γ := Set (Sieve (glueData hf).glued))
    ((grothendieckTopology IsOpenImmersion : (a : Scheme) → Set (Sieve a)) (glueData hf).glued)
    (Sieve.ofArrows (I :=
      PreZeroHypercover.I₀ (S := (glueData hf).glued)
        (Precoverage.ZeroHypercover.toPreZeroHypercover (glueData hf).openCover))
      (PreZeroHypercover.X (Precoverage.ZeroHypercover.toPreZeroHypercover (glueData hf).openCover))
      (PreZeroHypercover.f (Precoverage.ZeroHypercover.toPreZeroHypercover (glueData hf).openCover)))
refine :
  Sieve.pullback α
        (Sieve.ofArrows (I :=
          PreZeroHypercover.I₀ (S := (glueData hf).glued)
            (Precoverage.ZeroHypercover.toPreZeroHypercover (glueData hf).openCover))
          (PreZeroHypercover.X (Precoverage.ZeroHypercover.toPreZeroHypercover (glueData hf).openCover))
          (PreZeroHypercover.f (Precoverage.ZeroHypercover.toPreZeroHypercover (glueData hf).openCover))) ⊓
      Sieve.pullback β
        (Sieve.ofArrows (I :=
          PreZeroHypercover.I₀ (S := (glueData hf).glued)
            (Precoverage.ZeroHypercover.toPreZeroHypercover (glueData hf).openCover))
          (PreZeroHypercover.X (Precoverage.ZeroHypercover.toPreZeroHypercover (glueData hf).openCover))
          (PreZeroHypercover.f (Precoverage.ZeroHypercover.toPreZeroHypercover (glueData hf).openCover))) ≤
    Presheaf.equalizerSieve (FD := TypeCat.Fun) (F :=
      (zariskiTopology.yoneda.obj (glueData hf).glued).obj (P := Presheaf.IsSheaf zariskiTopology)) α β
⊢ Membership.mem (γ := Set (Sieve (unop (op U)))) ((zariskiTopology : (a : Scheme) → Set (Sieve a)) (unop (op U)))
    (Presheaf.equalizerSieve (FD := TypeCat.Fun) (F :=
      (zariskiTopology.yoneda.obj (glueData hf).glued).obj (P := Presheaf.IsSheaf zariskiTopology)) α β)

Tactic:
refine GrothendieckTopology.superset_covering _ ?_
      (zariskiTopology.intersection_covering (zariskiTopology.pullback_stable α mem)
        (zariskiTopology.pullback_stable β mem))

Premises used: CategoryTheory.GrothendieckTopology.superset_covering, AlgebraicGeometry.Scheme.zariskiTopology, CategoryTheory.GrothendieckTopology.pullback_stable, CategoryTheory.GrothendieckTopology.intersection_covering