Theorem 2elresin 2733

Description: Membership in two functions restricted by each other's domain.

Assertion

Ref	Expression
2elresin	⊢ ((F Fn A ∧ G Fn B) → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) ↔ (⟨x, y⟩ ∈ (F ↾ (A ∩ B)) ∧ ⟨x, z⟩ ∈ (G ↾ (A ∩ B)))))

Proof of Theorem 2elresin

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . 8 ⊢ x ∈ V
2		visset 1350	. . . . . . . 8 ⊢ y ∈ V
3	1, 2	fnop 2727	. . . . . . 7 ⊢ ((F Fn A ∧ ⟨x, y⟩ ∈ F) → x ∈ A)
4		visset 1350	. . . . . . . 8 ⊢ z ∈ V
5	1, 4	fnop 2727	. . . . . . 7 ⊢ ((G Fn B ∧ ⟨x, z⟩ ∈ G) → x ∈ B)
6	3, 5	anim12i 268	. . . . . 6 ⊢ (((F Fn A ∧ ⟨x, y⟩ ∈ F) ∧ (G Fn B ∧ ⟨x, z⟩ ∈ G)) → (x ∈ A ∧ x ∈ B))
7		an4 388	. . . . . 6 ⊢ (((F Fn A ∧ G Fn B) ∧ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) ↔ ((F Fn A ∧ ⟨x, y⟩ ∈ F) ∧ (G Fn B ∧ ⟨x, z⟩ ∈ G)))
8		elin 1635	. . . . . 6 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
9	6, 7, 8	3imtr4 192	. . . . 5 ⊢ (((F Fn A ∧ G Fn B) ∧ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) → x ∈ (A ∩ B))
10	2	opres 2580	. . . . . . 7 ⊢ (x ∈ (A ∩ B) → (⟨x, y⟩ ∈ (F ↾ (A ∩ B)) ↔ ⟨x, y⟩ ∈ F))
11	4	opres 2580	. . . . . . 7 ⊢ (x ∈ (A ∩ B) → (⟨x, z⟩ ∈ (G ↾ (A ∩ B)) ↔ ⟨x, z⟩ ∈ G))
12	10, 11	anbi12d 476	. . . . . 6 ⊢ (x ∈ (A ∩ B) → ((⟨x, y⟩ ∈ (F ↾ (A ∩ B)) ∧ ⟨x, z⟩ ∈ (G ↾ (A ∩ B))) ↔ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)))
13	12	biimprd 136	. . . . 5 ⊢ (x ∈ (A ∩ B) → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) → (⟨x, y⟩ ∈ (F ↾ (A ∩ B)) ∧ ⟨x, z⟩ ∈ (G ↾ (A ∩ B)))))
14	9, 13	syl 12	. . . 4 ⊢ (((F Fn A ∧ G Fn B) ∧ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) → (⟨x, y⟩ ∈ (F ↾ (A ∩ B)) ∧ ⟨x, z⟩ ∈ (G ↾ (A ∩ B)))))
15	14	exp 291	. . 3 ⊢ ((F Fn A ∧ G Fn B) → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) → (⟨x, y⟩ ∈ (F ↾ (A ∩ B)) ∧ ⟨x, z⟩ ∈ (G ↾ (A ∩ B))))))
16	15	pm2.43d 59	. 2 ⊢ ((F Fn A ∧ G Fn B) → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) → (⟨x, y⟩ ∈ (F ↾ (A ∩ B)) ∧ ⟨x, z⟩ ∈ (G ↾ (A ∩ B)))))
17		resss 2587	. . . . 5 ⊢ (F ↾ (A ∩ B)) ⊆ F
18	17	sseli 1504	. . . 4 ⊢ (⟨x, y⟩ ∈ (F ↾ (A ∩ B)) → ⟨x, y⟩ ∈ F)
19		resss 2587	. . . . 5 ⊢ (G ↾ (A ∩ B)) ⊆ G
20	19	sseli 1504	. . . 4 ⊢ (⟨x, z⟩ ∈ (G ↾ (A ∩ B)) → ⟨x, z⟩ ∈ G)
21	18, 20	anim12i 268	. . 3 ⊢ ((⟨x, y⟩ ∈ (F ↾ (A ∩ B)) ∧ ⟨x, z⟩ ∈ (G ↾ (A ∩ B))) → (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G))
22	21	a1i 7	. 2 ⊢ ((F Fn A ∧ G Fn B) → ((⟨x, y⟩ ∈ (F ↾ (A ∩ B)) ∧ ⟨x, z⟩ ∈ (G ↾ (A ∩ B))) → (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)))
23	16, 22	impbid 397	1 ⊢ ((F Fn A ∧ G Fn B) → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) ↔ (⟨x, y⟩ ∈ (F ↾ (A ∩ B)) ∧ ⟨x, z⟩ ∈ (G ↾ (A ∩ B)))))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∈ wcel 1092   ∩ cin 1486  ⟨cop 1810   ↾ cres 2412   Fn wfn 2417

This theorem is referenced by:  tfrlem5 2953

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-dm 2428  df-res 2430  df-fn 2433

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