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

Lemma filterlimi_locally : forall {F} {FF : Filter F} (f : T -> U -> Prop) y, filterlimi f F (locally y) <-> forall eps : posreal, F (fun x => exists z, f x z /\ ball y eps z).