-- Module: Mathlib.Topology.Sheaves.Points
Goal:
X : Type u
inst : TopologicalSpace X
x : X
U : Opens X
R : Sieve U
hR : R ∈ (grothendieckTopology X : (a : Opens X) → Set (Sieve a)) U
hU : x ∈ U
obtain :
  ∀ (V : Opens X) (f : V ⟶ U) (hf : R.arrows f) (hV : x ∈ V),
    ∃ (Y : Opens X) (f : Y ⟶ U) (_ : R.arrows f) (y :
      Functor.obj
        { obj := fun (U : Opens X) ↦ ULift.{u, 0} (PLift (x ∈ U)),
          map := fun {X_1 Y : Opens X} (f : X_1 ⟶ Y) ↦
            ↾fun (h : ULift.{u, 0} (PLift (x ∈ X_1))) ↦ { down := { down := ⋯ } },
          map_id := ⋯, map_comp := ⋯ }
        Y),
      (ConcreteCategory.hom (FC := TypeCat.Fun)
              (CategoryTheory.Functor.map
                { obj := fun (U : Opens X) ↦ ULift.{u, 0} (PLift (x ∈ U)),
                  map := fun {X_1 Y : Opens X} (f : X_1 ⟶ Y) ↦
                    ↾fun (h : ULift.{u, 0} (PLift (x ∈ X_1))) ↦
                      { down := { down := ⋯ : PLift (x ∈ Y) } : ULift.{u, 0} (PLift (x ∈ Y)) },
                  map_id := ⋯, map_comp := ⋯ : Opens X ⥤ Type u }
                f) :
            (a :
                (fun (X : Type u) ↦ X)
                  ({ obj := fun (U : Opens X) ↦ ULift.{u, 0} (PLift (x ∈ U)),
                        map := fun {X_1 Y : Opens X} (f : X_1 ⟶ Y) ↦
                          ↾fun (h : ULift.{u, 0} (PLift (x ∈ X_1))) ↦ { down := { down := ⋯ } },
                        map_id := ⋯, map_comp := ⋯ }.obj
                    Y)) →
              (fun (X : Type u) ↦ X)
                ({ obj := fun (U : Opens X) ↦ ULift.{u, 0} (PLift (x ∈ U)),
                      map := fun {X_1 Y : Opens X} (f : X_1 ⟶ Y) ↦
                        ↾fun (h : ULift.{u, 0} (PLift (x ∈ X_1))) ↦ { down := { down := ⋯ } },
                      map_id := ⋯, map_comp := ⋯ }.obj
                  U))
          y =
        { down := { down := hU : PLift (x ∈ U) } : ULift.{u, 0} (PLift (x ∈ U)) }
⊢ ∃ (Y : Opens X) (f : Y ⟶ U) (_ : R.arrows f) (y :
    Functor.obj
      { obj := fun (U : Opens X) ↦ ULift.{u, 0} (PLift (x ∈ U)),
        map := fun {X_1 Y : Opens X} (f : X_1 ⟶ Y) ↦
          ↾fun (h : ULift.{u, 0} (PLift (x ∈ X_1))) ↦ { down := { down := ⋯ } },
        map_id := ⋯, map_comp := ⋯ }
      Y),
    (ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              { obj := fun (U : Opens X) ↦ ULift.{u, 0} (PLift (x ∈ U)),
                map := fun {X_1 Y : Opens X} (f : X_1 ⟶ Y) ↦
                  ↾fun (h : ULift.{u, 0} (PLift (x ∈ X_1))) ↦
                    { down := { down := ⋯ : PLift (x ∈ Y) } : ULift.{u, 0} (PLift (x ∈ Y)) },
                map_id := ⋯, map_comp := ⋯ : Opens X ⥤ Type u }
              f) :
          (a :
              (fun (X : Type u) ↦ X)
                ({ obj := fun (U : Opens X) ↦ ULift.{u, 0} (PLift (x ∈ U)),
                      map := fun {X_1 Y : Opens X} (f : X_1 ⟶ Y) ↦
                        ↾fun (h : ULift.{u, 0} (PLift (x ∈ X_1))) ↦ { down := { down := ⋯ } },
                      map_id := ⋯, map_comp := ⋯ }.obj
                  Y)) →
            (fun (X : Type u) ↦ X)
              ({ obj := fun (U : Opens X) ↦ ULift.{u, 0} (PLift (x ∈ U)),
                    map := fun {X_1 Y : Opens X} (f : X_1 ⟶ Y) ↦
                      ↾fun (h : ULift.{u, 0} (PLift (x ∈ X_1))) ↦ { down := { down := ⋯ } },
                    map_id := ⋯, map_comp := ⋯ }.obj
                U))
        y =
      { down := { down := hU : PLift (x ∈ U) } : ULift.{u, 0} (PLift (x ∈ U)) }

Tactic:
obtain ⟨V, f, hf, hV⟩ := hR x hU