Section restricted_to_1.
Variable A F : Ens.
Section res_to_1a.
Variable X : Ens.
Hypothesis i : IN X (Domain (restricted_to A F)).
Let Y := EV (restricted_to A F) X.
Let stepa1 : IN (PAIR X Y) (restricted_to A F).
Proof EV_th1 (restricted_to A F) X i.
Let stepa2 : IN (PAIR X Y) F.
Proof Set_Of_th3 F (fun M : Ens => IN (PR1 M) A) (PAIR X Y) stepa1.
Let U := PAIR X Y.
Let stepa3 : IN (PR1 U) A.
Proof Set_Of_th2 F (fun M : Ens => IN (PR1 M) A) U stepa1.
Let stepa4 : EQ X (PR1 U).
Proof PAIR_proj_th3 X Y.
Let stepa5 : IN X A.
Proof Extensionality X (PR1 U) A stepa4 stepa3.
Let stepa6 : IN X (Domain F).
Proof Domain_th1 F X Y stepa2.
Lemma  (*Fact*) res_to_1a_Out : IN X (intersection A (Domain F)).
Proof intersection_th3 A (Domain F) X stepa5 stepa6.
End res_to_1a.
Let step1 : SUB (Domain (restricted_to A F)) (intersection A (Domain F)).
Proof res_to_1a_Out.
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.

Lemma res_to_1_Out :
EQ (intersection A (Domain F)) (Domain (restricted_to A F)).