Theorem brinxp 4792

Description: Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.)

Assertion

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

Proof of Theorem brinxp

Step	Hyp	Ref	Expression
1		brinxp2 4791	. . 3 |- ( A ( R i^i ( C X. D ) ) B <-> ( A e. C /\ B e. D /\ A R B ) )
2		df-3an 1004	. . 3 |- ( ( A e. C /\ B e. D /\ A R B ) <-> ( ( A e. C /\ B e. D ) /\ A R B ) )
3	1, 2	bitri 184	. 2 |- ( A ( R i^i ( C X. D ) ) B <-> ( ( A e. C /\ B e. D ) /\ A R B ) )
4	3	baibr 925	1 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> A ( R i^i ( C X. D ) ) B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   e. wcel 2200   i^i cin 3197   class class class wbr 4086   X. cxp 4721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729

This theorem is referenced by:  poinxp  4793  soinxp  4794  seinxp  4795  isores2  5949  ltpiord  7529