Proof using.
induction xs; intros.
- auto.
- simpl. rewrite H by intuition. rewrite IHxs by intuition. auto.
Qed.

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