induction ls; intuition; simpl in *.
f_equal.
eapply H.
intuition.
eapply IHls.
intuition.
Qed.

(* Specifications: *)
Theorem map_eq_subset :
forall (A B : Type)(ls2 ls1 : list A)(f1 f2 : A -> B),
map f1 ls1 = map f2 ls1 ->
(forall a, In a ls2 -> In a ls1) ->
map f1 ls2 = map f2 ls2.