(* End of proof of lem_exp_elem_prop_recip                                 *)

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(* Famille d'ensembles indexee                                             *)
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Definition index (a I : E) (funa : fun_ a) (doma : dom a = I) :=
  subset (fun x : E => True) (Img a).

(* Specifications: *)
Lemma lem_index_prop : forall (a I : E) (funa : fun_ a) (doma : dom a = I), index a I funa doma = Img a.
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).
Lemma lem_index_inter : forall (a I : E) (funa : fun_ a) (doma : dom a = I) (Ine : I <> Vide) (x : E) (indNe : index a I funa doma <> Vide), In x (Einter (index a I funa doma) indNe) -> forall i : E, In i I -> In x (App a funa i).