-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
h :
  Eq (α := Set (↑(↑(D.U j) : TopCat) : Type v))
    (Set.preimage (β := (↑(D.mapGlueData (forget C)).glued : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (Y := (D.mapGlueData (forget C)).glued) ((D.mapGlueData (forget C)).ι j)) :
        (a : (↑(↑(D.U j) : TopCat) : Type v)) → (↑(D.mapGlueData (forget C)).glued : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (Y := (D.mapGlueData (forget C)).glued) ((D.mapGlueData (forget C)).ι i)) :
          (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(D.mapGlueData (forget C)).glued : Type v)) ''
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
    (Set.image (α := (↑(↑(D.V (j, i)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.f j i).base) :
        (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.U j) : TopCat) : Type v))
      (Set.preimage (β := (↑(↑(D.V (i, j)) : TopCat) : Type v))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.t j i).base) :
          (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
        ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (D.f i j).base) :
            (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
          (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))))
h₁ :
  Function.Injective (α := (↑(D.mapGlueData (forget C)).glued : Type v)) (β := (↑(↑D.glued : TopCat) : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (D.mapGlueData (forget C)).glued) (Y := (↑D.glued : TopCat))
            (Iso.inv (D.gluedIso (forget C)) : (D.mapGlueData (forget C)).glued ⟶ Functor.obj (forget C) D.glued) :
          C((↑(D.mapGlueData (forget C)).glued : Type v), (↑(↑D.glued : TopCat) : Type v))) :
      (a : (↑(D.mapGlueData (forget C)).glued : Type v)) → (↑(↑D.glued : TopCat) : Type v))
⊢ Eq (α := Set (↑(↑(D.U j) : TopCat) : Type v))
    (Set.preimage (β := (↑(↑D.glued : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (Y := (↑D.glued : TopCat))
            (CategoryStruct.comp (Y := (D.mapGlueData (forget C)).glued)
              ((D.mapGlueData (forget C)).ι j : (D.mapGlueData (forget C)).U j ⟶ (D.mapGlueData (forget C)).glued)
              (Iso.inv (D.gluedIso (forget C)) : (D.mapGlueData (forget C)).glued ⟶ Functor.obj (forget C) D.glued))) :
        (a : (↑(↑(D.U j) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (Y := (↑D.glued : TopCat))
              (CategoryStruct.comp (Y := (D.mapGlueData (forget C)).glued)
                ((D.mapGlueData (forget C)).ι i : (D.mapGlueData (forget C)).U i ⟶ (D.mapGlueData (forget C)).glued)
                (Iso.inv (D.gluedIso (forget C)) :
                  (D.mapGlueData (forget C)).glued ⟶ Functor.obj (forget C) D.glued))) :
          (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v)) ''
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
    (Set.image (α := (↑(↑(D.V (j, i)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.f j i).base) :
        (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.U j) : TopCat) : Type v))
      (Set.preimage (β := (↑(↑(D.V (i, j)) : TopCat) : Type v))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.t j i).base) :
          (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
        ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (D.f i j).base) :
            (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
          (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))))

Tactic:
rw [TopCat.coe_comp, TopCat.coe_comp, Set.image_comp, Set.preimage_comp, Set.preimage_image_eq]

Premises used: Set.preimage_comp, Set.image_comp, Set.preimage_image_eq, TopCat.coe_comp