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Theorem brinxp 4712
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
StepHypRef Expression
1 brinxp2 4711 . . 3  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A  e.  C  /\  B  e.  D  /\  A R B ) )
2 df-3an 982 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
31, 2bitri 184 . 2  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
43baibr 921 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <-> 
A ( R  i^i  ( C  X.  D
) ) B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    e. wcel 2160    i^i cin 3143   class class class wbr 4018    X. cxp 4642
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650
This theorem is referenced by:  poinxp  4713  soinxp  4714  seinxp  4715  isores2  5834  ltpiord  7347
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