From Coq Require Export Ensembles.
Inductive characteristic_function_abstraction {X:Type} (P:X->Prop) (x:X) : Prop :=
| intro_characteristic_sat: P x ->
In (characteristic_function_abstraction P) x.
Definition characteristic_function_to_ensemble {X:Type} (P:X->Prop) : Ensemble X :=
characteristic_function_abstraction P.
Notation "[ x : X | P ]" :=
(characteristic_function_to_ensemble (fun x:X => P))
(x ident).

Lemma characteristic_function_to_ensemble_is_identity:
forall {X:Type} (P:X->Prop),
[ x:X | P x ] = P.