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

(* Specifications: *)
Lemma filterlimi_comp : forall T U V (f : T -> U) (g : U -> V -> Prop) F G H, filterlim f F G -> filterlimi g G H -> filterlimi (fun x => g (f x)) F H.
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.
Lemma filterlimi_ext : forall {T U F G} {FF : Filter F} (f g : T -> U -> Prop), (forall x y, f x y <-> g x y) -> filterlimi f F G -> filterlimi g F G.
Lemma filterlimi_filter_le_1 : forall {T U F G H} (f : T -> U -> Prop), filter_le G F -> filterlimi f F H -> filterlimi f G H.
Lemma filterlimi_filter_le_2 : forall {T U F G H} (f : T -> U -> Prop), filter_le G H -> filterlimi f F G -> filterlimi f F H.