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Theorem brinxp 4607
Description: Intersection of binary relation with cross 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 4606 . . 3  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A  e.  C  /\  B  e.  D  /\  A R B ) )
2 df-3an 964 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
31, 2bitri 183 . 2  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
43baibr 905 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 103    <-> wb 104    /\ w3a 962    e. wcel 1480    i^i cin 3070   class class class wbr 3929    X. cxp 4537
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545
This theorem is referenced by:  poinxp  4608  soinxp  4609  seinxp  4610  isores2  5714  ltpiord  7139
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