∀ {E : Type u_1} {X : Type u_2} [inst : TopologicalSpace E] [inst_1 : TopologicalSpace X] {f : E → X} (s : Set X)
  {I : Type u_3} [inst_2 : TopologicalSpace I],
  IsOpen s → IsOpen (f ⁻¹' s) → ∀ {x : X}, x ∈ s → IsEvenlyCovered (fun e => f ↑e) x I → IsEvenlyCovered f x I