Definition filterlimi {T U : Type} (f : T -> U -> Prop) F G :=
  filter_le (filtermapi f F) G.

Lemma filterlimi_ext_loc : forall {T U F G} {FF : Filter F} (f g : T -> U -> Prop), F (fun x => forall y, f x y <-> g x y) -> filterlimi f F G -> filterlimi g F G.