∀ {E : Type u_2} {X : Type u_1} [inst : TopologicalSpace E] [inst_1 : TopologicalSpace X] {f : E → X} {I : Type u_3}
  [inst_2 : TopologicalSpace I] {x : X} (U : Set X),
  (x ∈ U ∧ IsOpen U ∧ IsOpen (f ⁻¹' U) ∧ ∃ H, ∀ (x : ↑(f ⁻¹' U)), ↑(H x).1 = f ↑x) → x ∈ U