-- Lean 4 proof step from Mathlib.Topology.Gluing (goal)
Goal:
D : GlueData
i j : D.J
rw :
  Eq (α := Set (↑(D.U j) : Type u))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.range (ι := (↑(D.U i) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.ι i)) :
          (a : (↑(D.U i) : Type u)) → (↑D.glued : Type u))))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.image (α := (↑(D.U j) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.ι j)) :
          (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
        (Set.range (ι := (↑(D.V (j, i)) : Type u))
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                (D.f j i)) :
            (a : (↑(D.V (j, i)) : Type u)) → (↑(D.U j) : Type u)))))
rw₁ :
  Eq (α := Set (↑(D.U j) : Type u))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.image (α := (↑(D.U i) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.ι i)) :
          (a : (↑(D.U i) : Type u)) → (↑D.glued : Type u))
        Set.univ))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.image (α := (↑(D.U j) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.ι j)) :
          (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
        (Set.range (ι := (↑(D.V (j, i)) : Type u))
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                (D.f j i)) :
            (a : (↑(D.V (j, i)) : Type u)) → (↑(D.U j) : Type u)))))
rw₂ :
  Eq (α := Set (↑(D.U j) : Type u))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.image (α := (↑(D.U i) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.ι i)) :
          (a : (↑(D.U i) : Type u)) → (↑D.glued : Type u))
        Set.univ))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.image (α := (↑(D.U j) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.ι j)) :
          (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
        (Set.image (α := (↑(D.V (j, i)) : Type u))
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                (D.f j i)) :
            (a : (↑(D.V (j, i)) : Type u)) → (↑(D.U j) : Type u))
          Set.univ)))
rw₃ :
  Eq (α := Set (↑(D.U j) : Type u))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.image (α := (↑(D.U i) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.ι i)) :
          (a : (↑(D.U i) : Type u)) → (↑D.glued : Type u))
        Set.univ))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.image (α := (↑(D.V (j, i)) : Type u))
        (Function.comp (β := (↑(D.U j) : Type u))
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
                D.glued) (D.ι j)) :
            (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                (D.f j i)) :
            (a : (↑(D.V (j, i)) : Type u)) → (↑(D.U j) : Type u)))
        Set.univ))
rw₄ :
  Eq (α := Set (↑(D.U j) : Type u))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.image (α := (↑(D.U i) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.ι i)) :
          (a : (↑(D.U i) : Type u)) → (↑D.glued : Type u))
        Set.univ))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.image (α := (↑(D.V (j, i)) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.f j i ≫ D.ι j)) :
          (a : (↑(D.V (j, i)) : Type u)) → (↑D.glued : Type u))
        Set.univ))
rw₅ :
  Eq (α := Set (↑(D.U j) : Type u))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.range (ι := (↑(D.U i) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.ι i)) :
          (a : (↑(D.U i) : Type u)) → (↑D.glued : Type u))))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.image (α := (↑(D.V (j, i)) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.f j i ≫ D.ι j)) :
          (a : (↑(D.V (j, i)) : Type u)) → (↑D.glued : Type u))
        Set.univ))
rw₆ :
  Eq (α := Set (↑(D.U j) : Type u))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.range (ι := (↑(D.U i) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.ι i)) :
          (a : (↑(D.U i) : Type u)) → (↑D.glued : Type u))))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.range (ι := (↑(D.V (j, i)) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.f j i ≫ D.ι j)) :
          (a : (↑(D.V (j, i)) : Type u)) → (↑D.glued : Type u))))
rw₇ :
  Eq (α := Set (↑(D.U j) : Type u))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.range (ι := (↑(D.U i) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.ι i)) :
          (a : (↑(D.U i) : Type u)) → (↑D.glued : Type u))))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.range (ι := (↑(D.U j) : Type u))
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
                D.glued) (D.ι j)) :
            (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u)) ∩
        Set.range (ι := (↑(D.U i) : Type u))
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
                D.glued) (D.ι i)) :
            (a : (↑(D.U i) : Type u)) → (↑D.glued : Type u))))
⊢ Eq (α := Set (↑(D.U j) : Type u))
    (Set.preimage (β := (↑D.glued : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι j)) :
        (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
      (Set.range (ι := (↑(D.U i) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
              D.glued) (D.ι i)) :
          (a : (↑(D.U i) : Type u)) → (↑D.glued : Type u))))
    (Set.range (ι := (↑(D.V (j, i)) : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
            (D.f j i)) :
        (a : (↑(D.V (j, i)) : Type u)) → (↑(D.U j) : Type u)))
have rw₈ : Eq (α := Set (↑(D.U j) : Type u))
  (Set.preimage (β := (↑D.glued : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
          D.glued) (D.ι j)) :
      (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
    (Set.range (ι := (↑(D.U i) : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι i)) :
        (a : (↑(D.U i) : Type u)) → (↑D.glued : Type u))))
  (Set.preimage (β := (↑D.glued : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
          D.glued) (D.ι j)) :
      (a : (↑(D.U j) : Type u)) → (↑D.glued : Type u))
    (Set.range (ι := (↑(D.U i) : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued) (D.ι i)) :
        (a : (↑(D.U i) : Type u)) → (↑D.glued : Type u))))
Tactic: rw [← Set.preimage_image_eq (Set.range (D.f j i)) (D.ι_injective j), ← Set.image_univ, ←
    Set.image_univ, ← Set.image_comp, ← coe_comp, Set.image_univ, Set.image_univ, ← image_inter,
    Set.preimage_range_inter]