-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
U : TopologicalSpace.Opens ↥c.pt
x : ↥c.pt
hxU : x ∈ U
i : I
V : TopologicalSpace.Opens ↥(D.obj i)
hV : IsAffineOpen V
hxV : Membership.mem (α := ↥(D.obj i)) (↑V : Set ↥(D.obj i)) ((c.π.app i) x : ↥(D.obj i))
this_1 : ∀ (j : (Over i)ᵒᵖ), IsAffine.{u} (Opposite.unop ((opensDiagram D i V).op.obj j))
this : ∀ (j : Over i), IsAffine.{u} ((opensDiagram D i V).obj j)
r : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ V) : Type u)
hrU : (Functor.obj (Functor.obj (Functor.const I) c.pt) i).basicOpen r ≤ U
hxr :
  (↑(Subtype.mk (p := fun (x : ↥(Functor.obj (Functor.obj (Functor.const I) c.pt) i)) ↦ x ∈ U) x hxU) :
      ↥(((Functor.const I).obj c.pt).obj i)) ∈
    (Functor.obj (Functor.obj (Functor.const I) c.pt) i).basicOpen r
j : Over i
s :
  ((opensDiagram D i V).op ⋙
        Scheme.Γ ⋙
          forget CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier)).obj
    (Opposite.op j)
hs :
  Eq (α :=
    (fun (X : Type u) ↦ X)
      (Functor.obj
        (Functor.obj (Functor.const (Over i)ᵒᵖ)
          ((Scheme.Γ ⋙
                  forget CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                    CommRingCat.carrier)).mapCocone
              (opensCone D c i V).op).pt)
        (Opposite.op j)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (((Scheme.Γ ⋙
                        forget CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                          CommRingCat.carrier)).mapCocone
                    (opensCone D c i V).op).ι.app
              (Opposite.op j)) :
          (a : (fun (X : Type u) ↦ X) (((opensDiagram D i V).op ⋙ Scheme.Γ ⋙ forget CommRingCat).obj (Opposite.op j))) →
            (fun (X : Type u) ↦ X)
              (((Functor.const (Over i)ᵒᵖ).obj
                    ((Scheme.Γ ⋙ forget CommRingCat).mapCocone (opensCone D c i V).op).pt).obj
                (Opposite.op j)))
        s :
      (fun (X : Type u) ↦ X)
        (Functor.obj
          (Functor.obj (Functor.const (Over i)ᵒᵖ)
            ((Scheme.Γ ⋙
                    forget CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                      CommRingCat.carrier)).mapCocone
                (opensCone D c i V).op).pt)
          (Opposite.op j)))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (c.π.app i ⁻¹ᵁ V).topIso.inv :
          (a : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ V) : Type u)) →
            (↑Γ((↑(c.π.app i ⁻¹ᵁ V) : Scheme), ⊤) : Type u))
        r :
      (↑Γ((↑(c.π.app i ⁻¹ᵁ V) : Scheme), ⊤) : Type u))
obtain :
  ∀ (s : (↑Γ(D.obj j.left, D.map j.hom ⁻¹ᵁ V) : Type u))
    (hs :
      Eq (α :=
        (fun (X : Type u) ↦ X)
          (Functor.obj
            (Functor.obj (Functor.const (Over i)ᵒᵖ)
              ((Scheme.Γ ⋙
                      forget CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                        CommRingCat.carrier)).mapCocone
                  (opensCone D c i V).op).pt)
            (Opposite.op j)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (((Scheme.Γ ⋙
                            forget CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u))
                              (CC := CommRingCat.carrier)).mapCocone
                        (opensCone D c i V).op).ι.app
                  (Opposite.op j)) :
              (a :
                  (fun (X : Type u) ↦ X)
                    (((opensDiagram D i V).op ⋙ Scheme.Γ ⋙ forget CommRingCat).obj (Opposite.op j))) →
                (fun (X : Type u) ↦ X)
                  (((Functor.const (Over i)ᵒᵖ).obj
                        ((Scheme.Γ ⋙ forget CommRingCat).mapCocone (opensCone D c i V).op).pt).obj
                    (Opposite.op j)))
            (((D.map j.hom ⁻¹ᵁ V).topIso.symm.commRingCatIsoToRingEquiv :
                (a : (↑Γ(D.obj j.left, D.map j.hom ⁻¹ᵁ V) : Type u)) →
                  (↑Γ((↑(D.map j.hom ⁻¹ᵁ V) : Scheme), ⊤) : Type u))
              s) :
          (fun (X : Type u) ↦ X)
            (Functor.obj
              (Functor.obj (Functor.const (Over i)ᵒᵖ)
                ((Scheme.Γ ⋙
                        forget CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                          CommRingCat.carrier)).mapCocone
                    (opensCone D c i V).op).pt)
              (Opposite.op j)))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (c.π.app i ⁻¹ᵁ V).topIso.inv :
              (a : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ V) : Type u)) →
                (↑Γ((↑(c.π.app i ⁻¹ᵁ V) : Scheme), ⊤) : Type u))
            r :
          (↑Γ((↑(c.π.app i ⁻¹ᵁ V) : Scheme), ⊤) : Type u))),
    ∃ (U' : TopologicalSpace.Opens ↥c.pt),
      Membership.mem (γ := Set (TopologicalSpace.Opens ↥c.pt))
          {x : TopologicalSpace.Opens ↥c.pt | ∃ (i : I) (V : (D.obj i).Opens) (_ : IsAffineOpen V), c.π.app i ⁻¹ᵁ V = x}
          U' ∧
        x ∈ U' ∧ U' ≤ U
⊢ ∃ (U' : TopologicalSpace.Opens ↥c.pt),
    Membership.mem (γ := Set (TopologicalSpace.Opens ↥c.pt))
        {x : TopologicalSpace.Opens ↥c.pt | ∃ (i : I) (V : (D.obj i).Opens) (_ : IsAffineOpen V), c.π.app i ⁻¹ᵁ V = x}
        U' ∧
      x ∈ U' ∧ U' ≤ U

Tactic:
obtain ⟨s, rfl⟩ := (D.map j.hom ⁻¹ᵁ V).topIso.symm.commRingCatIsoToRingEquiv.surjective s

Premises used: CategoryTheory.Iso.commRingCatIsoToRingEquiv, RingEquiv.surjective, AlgebraicGeometry.Scheme.Opens.topIso, AlgebraicGeometry.LocallyRingedSpace.Hom.toHom, AlgebraicGeometry.Scheme.Hom.toLRSHom', CategoryTheory.Functor.map, CategoryTheory.Iso.symm, AlgebraicGeometry.PresheafedSpace.Hom.base, TopologicalSpace.Opens.map, CategoryTheory.Over.hom, CategoryTheory.Functor.obj, rfl