-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
i j k : D.J
U : Opens (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)
this :
  Eq (α := (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v) → (↑(↑(D.V (i, k)) : TopCat) : Type v))
    (⇑(TopCat.Hom.hom (Hom.base (pullback.snd (D.f i j) (D.f i k)))) :
      (a : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)) → (↑(↑(D.V (i, k)) : TopCat) : Type v))
    (Function.comp (β := (↑(↑(D.V (k, i)) : TopCat) : Type v))
      (⇑(TopCat.Hom.hom (D.t k i).base) :
        (a : (↑(↑(D.V (k, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, k)) : TopCat) : Type v))
      (Function.comp (β := (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v))
        (⇑(TopCat.Hom.hom (Hom.base (pullback.fst (D.f k i) (D.f k j)))) :
          (a : (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v)) → (↑(↑(D.V (k, i)) : TopCat) : Type v))
        (⇑(TopCat.Hom.hom (inv (D.t' k i j)).base) :
          (a : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)) →
            (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v))))
w :
  Eq (α := Set (↑(↑(D.V (k, i)) : TopCat) : Type v))
    (Set.image (α := (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v))
      (⇑(TopCat.Hom.hom (Hom.base (pullback.fst (D.f k i) (D.f k j)))) :
        (a : (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v)) → (↑(↑(D.V (k, i)) : TopCat) : Type v))
      ((⇑(TopCat.Hom.hom (inv (D.t' k i j)).base) :
          (a : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)) →
            (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v)) ''
        (↑U : Set (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v))))
    (Set.image (α := (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (↑(pullback (D.f k i) (D.f k j)) : TopCat)) (Hom.base (pullback.fst (D.f k i) (D.f k j)))) :
        (a : (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v)) → (↑(↑(D.V (k, i)) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.t' k i j).base) :
          (a : (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v)) →
            (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)) ⁻¹'
        (↑U : Set (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v))))
w₁ :
  Function.Injective (α := (↑(↑(D.V (k, i)) : TopCat) : Type v)) (β := (↑(↑(D.V (i, k)) : TopCat) : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
          (D.t k i).base) :
      (a : (↑(↑(D.V (k, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, k)) : TopCat) : Type v))
⊢ Eq (α := Set (↑(↑(D.V (k, i)) : TopCat) : Type v))
    (Set.preimage (β := (↑(↑(D.V (i, k)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.t k i).base) :
        (a : (↑(↑(D.V (k, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, k)) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (X := (↑(pullback (D.f i j) (D.f i k)) : TopCat)) (Hom.base (pullback.snd (D.f i j) (D.f i k)))) :
          (a : (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)) → (↑(↑(D.V (i, k)) : TopCat) : Type v)) ''
        (↑U : Set (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v))))
    (Set.image (α := (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (↑(pullback (D.f k i) (D.f k j)) : TopCat)) (Hom.base (pullback.fst (D.f k i) (D.f k j)))) :
        (a : (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v)) → (↑(↑(D.V (k, i)) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.t' k i j).base) :
          (a : (↑(↑(pullback (D.f k i) (D.f k j)) : TopCat) : Type v)) →
            (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v)) ⁻¹'
        (↑U : Set (↑(↑(pullback (D.f i j) (D.f i k)) : TopCat) : Type v))))

Tactic:
rw [this, Set.image_comp, Set.image_comp, Set.preimage_image_eq]

Premises used: Set.image_comp, Set.preimage_image_eq