Section Image.
Variables U V : Type.
Inductive Im (X:Ensemble U) (f:U -> V) : Ensemble V :=
Im_intro : forall x:U, In _ X x -> forall y:V, y = f x -> In _ (Im X f) y.

Lemma Im_inv :
forall (X:Ensemble U) (f:U -> V) (y:V),
In _ (Im X f) y ->  exists x : U, In _ X x /\ f x = y.