∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] [T2Space Y]
  [CompactlyCoherentSpace Y] {f : X → Y},
  IsProperMap f ↔ Continuous f ∧ ∀ ⦃K : Set Y⦄, IsCompact K → IsCompact (f ⁻¹' K)