Lemma fin_subset_match {A B: Type} {P: A -> B -> Prop}: forall (X: Ensemble A) (Y: Ensemble B), (forall x, X x -> exists y, P x y /\ Y y) -> forall xs, Forall (fun x => X x) xs -> exists ys, Forall2 P xs ys /\ Forall (fun y => Y y) ys.