Theorem brinxp 2466

Description: Intersection of binary relation with cross product.

Assertion

Ref	Expression
brinxp	⊢ ((A ∈ C ∧ B ∈ D) → (ARB ↔ A(R ∩ (C × D))B))

Proof of Theorem brinxp

Step	Hyp	Ref	Expression
1		ibar 487	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → (ARB ↔ ((A ∈ C ∧ B ∈ D) ∧ ARB)))
2		opelxpi 2455	. . . . 5 ⊢ ((A ∈ C ∧ B ∈ D) → ⟨A, B⟩ ∈ (C × D))
3		ibib 448	. . . . 5 ⊢ (((A ∈ C ∧ B ∈ D) → ⟨A, B⟩ ∈ (C × D)) ↔ ((A ∈ C ∧ B ∈ D) → ((A ∈ C ∧ B ∈ D) ↔ ⟨A, B⟩ ∈ (C × D))))
4	2, 3	mpbi 164	. . . 4 ⊢ ((A ∈ C ∧ B ∈ D) → ((A ∈ C ∧ B ∈ D) ↔ ⟨A, B⟩ ∈ (C × D)))
5		df-br 2063	. . . . 5 ⊢ (ARB ↔ ⟨A, B⟩ ∈ R)
6	5	a1i 7	. . . 4 ⊢ ((A ∈ C ∧ B ∈ D) → (ARB ↔ ⟨A, B⟩ ∈ R))
7	4, 6	anbi12d 476	. . 3 ⊢ ((A ∈ C ∧ B ∈ D) → (((A ∈ C ∧ B ∈ D) ∧ ARB) ↔ (⟨A, B⟩ ∈ (C × D) ∧ ⟨A, B⟩ ∈ R)))
8		df-br 2063	. . . 4 ⊢ (A(R ∩ (C × D))B ↔ ⟨A, B⟩ ∈ (R ∩ (C × D)))
9		elin 1635	. . . 4 ⊢ (⟨A, B⟩ ∈ (R ∩ (C × D)) ↔ (⟨A, B⟩ ∈ R ∧ ⟨A, B⟩ ∈ (C × D)))
10		ancom 333	. . . 4 ⊢ ((⟨A, B⟩ ∈ R ∧ ⟨A, B⟩ ∈ (C × D)) ↔ (⟨A, B⟩ ∈ (C × D) ∧ ⟨A, B⟩ ∈ R))
11	8, 9, 10	3bitr 155	. . 3 ⊢ (A(R ∩ (C × D))B ↔ (⟨A, B⟩ ∈ (C × D) ∧ ⟨A, B⟩ ∈ R))
12	7, 11	syl6bbr 416	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → (((A ∈ C ∧ B ∈ D) ∧ ARB) ↔ A(R ∩ (C × D))B))
13	1, 12	bitrd 406	1 ⊢ ((A ∈ C ∧ B ∈ D) → (ARB ↔ A(R ∩ (C × D))B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∈ wcel 1092   ∩ cin 1486  ⟨cop 1810   class class class wbr 2054   × cxp 2408

This theorem is referenced by:  weinxp 2467

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

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