induction ls; intuition; simpl in *.
intuition; subst.
eapply H; eauto.
eapply IHls; eauto.
Qed.

Lemma rel_map_NoDup : forall (A B : Type)(lsa : list A)(lsb : list B) rel,
rel_map rel lsa lsb ->
NoDup lsa ->
(forall a1 a2 b1 b2, In a1 lsa -> In a2 lsa -> a1 <> a2 -> rel a1 b1 -> rel a2 b2 -> b1 <> b2) ->
NoDup lsb.