induction ls; intuition; simpl in *.
exfalso.
eapply H.
econstructor.
inversion H2; clear H2; subst.
destruct (in_dec (EqDec_dec eqd) a ls); intuition.
destruct (in_dec (EqDec_dec eqd) a ls); intuition.
eapply H0.
intuition.
eapply H1.
econstructor; intuition.
Qed.

Theorem hasDups_inj_equiv :
forall (A B : Set)(eqda : EqDec A)(eqdb : EqDec B)(lsa : list A)(inj : A -> B),
(forall a1 a2, inj a1 = inj a2 -> a1 = a2) ->
hasDups _ lsa = hasDups _ (map inj lsa).