Proof using.
intuition eauto using NoDup_Permutation_NoDup, Permutation_sym, Permutation_cons_append.
Qed.

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