Definition is_filter_lim (F : (T -> Prop) -> Prop) (x : T) :=
  forall P, locally x P -> F P.

Lemma filterlimi_locally_close : forall {F} {FF: ProperFilter F} (f : T -> U -> Prop) l l', F (fun x => forall y1 y2, f x y1 -> f x y2 -> y1 = y2) -> filterlimi f F (locally l) -> filterlimi f F (locally l') -> close l l'.