Proof using.
induction l; intros; simpl.
- auto.
- apply NoDup_append.
invc_NoDup.
constructor; auto.
intuition.
find_apply_lem_hyp in_rev.
auto.
Qed.

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