-- Module: Mathlib.AlgebraicGeometry.Pullbacks
Goal:
X Y Z : Scheme
𝒰_1 : X.OpenCover
f_1 : X ⟶ Z
g_1 : Y ⟶ Z
inst : ∀ (i : 𝒰_1.I₀), HasPullback (𝒰_1.f i ≫ f_1) g_1
s : PullbackCone f_1 g_1
𝒰 : OpenCover.{v, u} X
f : X ⟶ Z
g : Y ⟶ Z
x : ↥(pullback f g)
𝒱 : Cover (precoverage IsOpenImmersion) (limit (cospan f g)) :=
  Cover.copy (P := IsOpenImmersion)
    (Cover.pushforwardIso (P := IsOpenImmersion) (gluing (Cover.ulift (P := IsOpenImmersion) 𝒰) f g).openCover
      (Iso.inv
        (limit.isoLimitCone
          {
            cone :=
              PullbackCone.mk (W := (gluing (Cover.ulift (P := IsOpenImmersion) 𝒰) f g).glued)
                (p1 (Cover.ulift (P := IsOpenImmersion) 𝒰) f g) (p2 (Cover.ulift (P := IsOpenImmersion) 𝒰) f g) ⋯,
            isLimit := gluedIsLimit (Cover.ulift (P := IsOpenImmersion) 𝒰) f g : LimitCone (cospan f g) })))
    (Cover.ulift (P := IsOpenImmersion) 𝒰).I₀
    (fun (i : (Cover.ulift (P := IsOpenImmersion) 𝒰).I₀) ↦ pullback ((Cover.ulift (P := IsOpenImmersion) 𝒰).f i ≫ f) g)
    (fun (i : (Cover.ulift (P := IsOpenImmersion) 𝒰).I₀) ↦
      pullback.map ((Cover.ulift (P := IsOpenImmersion) 𝒰).f i ≫ f) g f g ((Cover.ulift (P := IsOpenImmersion) 𝒰).f i)
        (𝟙 Y) (𝟙 Z) ⋯ ⋯)
    (Equiv.refl (Cover.ulift (P := IsOpenImmersion) 𝒰).I₀)
    (fun (x : (Cover.ulift (P := IsOpenImmersion) 𝒰).I₀) ↦
      Iso.refl
        ((fun (i : (Cover.ulift (P := IsOpenImmersion) 𝒰).I₀) ↦
            pullback ((Cover.ulift (P := IsOpenImmersion) 𝒰).f i ≫ f) g)
          x))
    ⋯
refine_1 :
  ∀ (i : 𝒱.I₀) (x : ↥(𝒱.X i)),
    ∃ (i_1 :
      PreZeroHypercover.I₀.{v, u, u + 1} (C := Scheme) (S := pullback f g)
        { I₀ := 𝒰.I₀, X := fun (i : 𝒰.I₀) ↦ pullback (𝒰.f i ≫ f) g,
          f := fun (i : 𝒰.I₀) ↦ pullback.map (𝒰.f i ≫ f) g f g (𝒰.f i) (𝟙 Y) (𝟙 Z) ⋯ ⋯ }),
      Membership.mem (α := ↥(pullback f g)) (γ := Set ↥(pullback f g))
        (Set.range (ι :=
          ↥(PreZeroHypercover.X
              { I₀ := 𝒰.I₀, X := fun (i : 𝒰.I₀) ↦ pullback (𝒰.f i ≫ f) g,
                f := fun (i : 𝒰.I₀) ↦ pullback.map (𝒰.f i ≫ f) g f g (𝒰.f i) (𝟙 Y) (𝟙 Z) ⋯ ⋯ :
                PreZeroHypercover.{v, u, u + 1} (pullback f g) }
              i_1))
          ⇑(PreZeroHypercover.f
              { I₀ := 𝒰.I₀, X := fun (i : 𝒰.I₀) ↦ pullback (𝒰.f i ≫ f) g,
                f := fun (i : 𝒰.I₀) ↦ pullback.map (𝒰.f i ≫ f) g f g (𝒰.f i) (𝟙 Y) (𝟙 Z) ⋯ ⋯ :
                PreZeroHypercover.{v, u, u + 1} (pullback f g) }
              i_1))
        ((𝒱.f i) x : ↥(limit (cospan f g)))
⊢ ∃ (i :
    PreZeroHypercover.I₀.{v, u, u + 1} (C := Scheme) (S := pullback f g)
      { I₀ := 𝒰.I₀, X := fun (i : 𝒰.I₀) ↦ pullback (𝒰.f i ≫ f) g,
        f := fun (i : 𝒰.I₀) ↦ pullback.map (𝒰.f i ≫ f) g f g (𝒰.f i) (𝟙 Y) (𝟙 Z) ⋯ ⋯ }),
    Membership.mem (γ := Set ↥(pullback f g))
      (Set.range (ι :=
        ↥(PreZeroHypercover.X
            { I₀ := 𝒰.I₀, X := fun (i : 𝒰.I₀) ↦ pullback (𝒰.f i ≫ f) g,
              f := fun (i : 𝒰.I₀) ↦ pullback.map (𝒰.f i ≫ f) g f g (𝒰.f i) (𝟙 Y) (𝟙 Z) ⋯ ⋯ :
              PreZeroHypercover.{v, u, u + 1} (pullback f g) }
            i))
        ⇑(PreZeroHypercover.f
            { I₀ := 𝒰.I₀, X := fun (i : 𝒰.I₀) ↦ pullback (𝒰.f i ≫ f) g,
              f := fun (i : 𝒰.I₀) ↦ pullback.map (𝒰.f i ≫ f) g f g (𝒰.f i) (𝟙 Y) (𝟙 Z) ⋯ ⋯ :
              PreZeroHypercover.{v, u, u + 1} (pullback f g) }
            i))
      x

Tactic:
obtain ⟨i, x, rfl⟩ := 𝒱.exists_eq x

Premises used: AlgebraicGeometry.Scheme.Cover.exists_eq, rfl