induction 1; intuition; simpl in *.
f_equal.
eauto.
Qed.

Lemma rel_map_unary_pred : forall (A B : Type)(P : A -> B -> Prop)(lsa : list A)(lsb : list B)(P' : B -> Prop),
rel_map P lsa lsb ->
(forall a b, P a b -> P' b) ->
forall b, In b lsb -> P' b.