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_def :
forall (X:Ensemble U) (f:U -> V) (x:U), In _ X x -> In _ (Im X f) (f x).