(* Proof of lem_index_prop                                                 *)
intros; apply axs_extensionnalite; intros; unfold iff in |- *; split; intros.
unfold index in H.
elim (axs_comprehension (fun _ : E => True) (Img a) v2); intros;
generalize (H0 H); clear H H0 H1; intros; elim H;
clear H; intros; auto with zfc.
unfold index in |- *.
elim (axs_comprehension (fun _ : E => True) (Img a) v2); intros; apply H1;
clear H0 H1; intros.
split; auto with zfc.
Qed.
(* End of proof of lem_index_prop                                          *)

Lemma lem_reunion_index_prop :
forall (a I : E) (funa : fun_ a) (doma : dom a = I) (x : E),
In x (reunion (index a I funa doma)) ->
exists i : E, In i I /\ In x (App a funa i).