intuition.
eapply NoDup_app in H.
intuition.
Qed.

Theorem NoDup_map :
forall (A B : Type)(f : A -> B)(ls : list A),
NoDup (map f ls) ->
(NoDup ls /\ (forall b1 b2, In b1 ls -> In b2 ls -> f b1 = f b2 -> b1 = b2)).