Proof with auto.
induction l; simpl...
intros.
inversion_clear H0.
apply NoDup_cons...
intro.
destruct (In_map_inv H0).
destruct H3.
apply (H x a)...
intro.
subst...
Qed. (* todo: replace NoDup_map *)

Lemma NoDup_map A B (f: A -> B) l:
(forall x y, In x l -> In y l -> f x = f y -> x = y) -> NoDup l -> NoDup (map f l).