Theorem brinxp 4800

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 4799	. . 3 |- ( A ( R i^i ( C X. D ) ) B <-> ( A e. C /\ B e. D /\ A R B ) )
2		df-3an 1007	. . 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 928	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 1005   e. wcel 2202   i^i cin 3200   class class class wbr 4093   X. cxp 4729

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737

This theorem is referenced by:  poinxp  4801  soinxp  4802  seinxp  4803  isores2  5964  ltpiord  7582