Theorem brinxp2 4655

Description: Intersection of binary relation with Cartesian 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 4018	. 2 |- ( A ( R i^i ( C X. D ) ) B <-> ( A R B /\ A ( C X. D ) B ) )
2		ancom 264	. 2 |- ( ( A R B /\ A ( C X. D ) B ) <-> ( A ( C X. D ) B /\ A R B ) )
3		brxp 4619	. . . 4 |- ( A ( C X. D ) B <-> ( A e. C /\ B e. D ) )
4	3	anbi1i 454	. . 3 |- ( ( A ( C X. D ) B /\ A R B ) <-> ( ( A e. C /\ B e. D ) /\ A R B ) )
5		df-3an 965	. . 3 |- ( ( A e. C /\ B e. D /\ A R B ) <-> ( ( A e. C /\ B e. D ) /\ A R B ) )
6	4, 5	bitr4i 186	. 2 |- ( ( A ( C X. D ) B /\ A R B ) <-> ( A e. C /\ B e. D /\ A R B ) )
7	1, 2, 6	3bitri 205	1 |- ( A ( R i^i ( C X. D ) ) B <-> ( A e. C /\ B e. D /\ A R B ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 963   e. wcel 2128   i^i cin 3101   class class class wbr 3967   X. cxp 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-xp 4594

This theorem is referenced by:  brinxp  4656  fncnv  5238  erinxp  6556  isstructim  12274  isstructr  12275