induction 1; intuition; simpl in *.
econstructor.
inversion H1; clear H1; subst.
econstructor.
eapply ne_all_not_in.
intuition.
subst.
destruct (rel_map_inverse H a').
trivial.
intuition.
eapply H2.
left; eauto.
right; eauto.
intuition; subst; intuition.
eauto.
eauto.
trivial.
eapply IHrel_map; intuition.
subst.
eapply H2.
right.
eapply H3.
right.
eapply H4.
trivial.
eauto.
eauto.
trivial.
Qed.

Lemma rel_map_in : forall (A B : Type)(lsa : list A)(lsb : list B) rel,
rel_map rel lsa lsb ->
(forall a b1 b2, In a lsa -> rel a b1 -> rel a b2 -> b1 = b2) ->
forall b a,
In a lsa ->
rel a b ->
In b lsb.