Section res_to_1b.
Variable X : Ens.
Hypothesis i : IN X (intersection A (Domain F)).
Let stepb1 : IN X A.
Proof intersection_th1 A (Domain F) X i.
Let stepb2 : IN X (Domain F).
Proof intersection_th2 A (Domain F) X i.
Let Y := EV F X.
Let U := PAIR X Y.
Let stepb3 : IN U F.
Proof EV_th1 F X stepb2.
Let stepb4 : EQ X (PR1 U).
Proof PAIR_proj_th3 X Y.
Let stepb5 : EQ (PR1 U) X.
Proof EQ_symm X (PR1 U) stepb4.
Let stepb6 : IN (PR1 U) A.
Proof Extensionality (PR1 U) X A stepb5 stepb1.
Let stepb7 : IN U (restricted_to A F).
Proof restricted_to_th1 A F U stepb3 stepb6.
Lemma  (*Fact*) res_to_1b_Out : IN X (Domain (restricted_to A F)).
Proof Domain_th1 (restricted_to A F) X Y stepb7.
End res_to_1b.
Let step2 : SUB (intersection A (Domain F)) (Domain (restricted_to A F)).
Proof res_to_1b_Out.
Proof
Build_EQ (intersection A (Domain F)) (Domain (restricted_to A F)) step2
step1.
End restricted_to_1.

Theorem restricted_to_th5 :
forall A F : Ens,
EQ (intersection A (Domain F)) (Domain (restricted_to A F)).