Definition closed (D : T -> Prop) :=
  forall x, not (locally x (fun x : T => not (D x))) -> D x.

Lemma closed_filterlim_loc : forall {T} {U : UniformSpace} {F} {FF : ProperFilter' F} (f : T -> U) (D : U -> Prop), forall y, filterlim f F (locally y) -> F (fun x => D (f x)) -> closed D -> D y.