Theorem brinxp2 4473

Description: Intersection of binary relation with cross product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
brinxp2	|- ( A ( R i^i ( C X. D ) ) B <-> ( A e. C /\ B e. D /\ A R B ) )

Proof of Theorem brinxp2

Step	Hyp	Ref	Expression
1		brin 3867	. 2 |- ( A ( R i^i ( C X. D ) ) B <-> ( A R B /\ A ( C X. D ) B ) )
2		ancom 262	. 2 |- ( ( A R B /\ A ( C X. D ) B ) <-> ( A ( C X. D ) B /\ A R B ) )
3		brxp 4441	. . . 4 |- ( A ( C X. D ) B <-> ( A e. C /\ B e. D ) )
4	3	anbi1i 446	. . 3 |- ( ( A ( C X. D ) B /\ A R B ) <-> ( ( A e. C /\ B e. D ) /\ A R B ) )
5		df-3an 924	. . 3 |- ( ( A e. C /\ B e. D /\ A R B ) <-> ( ( A e. C /\ B e. D ) /\ A R B ) )
6	4, 5	bitr4i 185	. 2 |- ( ( A ( C X. D ) B /\ A R B ) <-> ( A e. C /\ B e. D /\ A R B ) )
7	1, 2, 6	3bitri 204	1 |- ( A ( R i^i ( C X. D ) ) B <-> ( A e. C /\ B e. D /\ A R B ) )

Syntax hints:   /\ wa 102   <-> wb 103   /\ w3a 922   e. wcel 1436   i^i cin 2987   class class class wbr 3820   X. cxp 4409

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-opab 3875  df-xp 4417

This theorem is referenced by:  brinxp  4474  fncnv  5045  erinxp  6318