Section Images1.
Variable E F : Setoid.
Variable f : MAP E F.
Definition image : part_set E -> part_set F.
intros A.
apply
(Build_Predicate
(Pred_fun:=fun y : F => exists x : E, in_part x A /\ Equal y (f x))).
red in |- *.
intros y y' H' H'0; try assumption.
elim H'; intros x E0; elim E0; intros H'1 H'2; try exact H'1; clear E0 H'.
exists x; split; [ try assumption | idtac ].
apply Trans with y; auto with algebra.
Defined.
simpl in |- *; auto with algebra.
Qed.
simpl in |- *; auto with algebra.
intros A x y H' H'0; try assumption.
exists x; split; [ try assumption | idtac ]; auto with algebra.
Qed.
Hint Resolve in_image: algebra.
intros A B H'; try assumption.
unfold included in |- *.
intros x H'0; try assumption.
elim H'0.
intros x0 H'1; try assumption.
apply in_image with (x := x0); auto with algebra.
red in H'.
elim H'1.
auto with algebra.
elim H'1; auto with algebra.
Qed.
Hint Resolve image_included: algebra.
intros A B H'; try assumption.
apply included_antisym; auto with algebra.
Qed.
Hint Resolve image_comp: algebra.

Lemma image_in_image :
forall (A : part_set E) (x : E), in_part x A -> in_part (f x) (image A).