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

Lemma is_filter_lim_filtermap {T: UniformSpace} {U : UniformSpace} : forall F x (f : T -> U), filterlim f (locally x) (locally (f x)) -> is_filter_lim F x -> is_filter_lim (filtermap f F) (f x).