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Theorem brinxp 4743
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 4742 . . 3  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A  e.  C  /\  B  e.  D  /\  A R B ) )
2 df-3an 983 . . 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 922 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 981    e. wcel 2176    i^i cin 3165   class class class wbr 4044    X. cxp 4673
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681
This theorem is referenced by:  poinxp  4744  soinxp  4745  seinxp  4746  isores2  5882  ltpiord  7432
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