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.