(* Proof of lem_fun_img_inv_is_in_pdom                                     *)
intros.
elim (axs_parties (dom f) (fun_img_inverse f Af x)); intros; apply H1;
clear H0 H1; intros.
unfold fun_img_inverse in H0.
elim
(axs_comprehension
(fun x0 : E => In x (parties (Img f)) -> In (App f Af x0) x)
(dom f) v3); intros; generalize (H1 H0); clear H0 H1 H2;
intros.
elim H0; clear H0; intros; auto with zfc.
Qed.
(* End of proof of lem_fun_img_inv_is_in_pdom                              *)

(* Specifications: *)
Lemma lem_fun_img_prop1 :
forall (f : E) (Af : fun_ f) (x : E),
In x (parties (Img f)) -> fun_img_directe f Af (fun_img_inverse f Af x) = x.