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Theorem brinxp 5012
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 5011 . . 3  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A  e.  C  /\  B  e.  D  /\  A R B ) )
2 df-3an 967 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
31, 2bitri 249 . 2  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
43baibr 897 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <-> 
A ( R  i^i  ( C  X.  D
) ) B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758    i^i cin 3438   class class class wbr 4403    X. cxp 4949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-xp 4957
This theorem is referenced by:  poinxp  5013  soinxp  5014  frinxp  5015  seinxp  5016  exfo  5973  isores2  6136  ltpiord  9170  ordpinq  9226  pwsleval  14553  tsrss  15515  ordtrest  18941  ordtrest2lem  18942  ordtrestNEW  26516  ordtrest2NEWlem  26517
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