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Theorem brinxp 4916
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 4915 . . 3  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A  e.  C  /\  B  e.  D  /\  A R B ) )
2 df-3an 984 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
31, 2bitri 252 . 2  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
43baibr 912 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 187    /\ wa 370    /\ w3a 982    e. wcel 1872    i^i cin 3435   class class class wbr 4423    X. cxp 4851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-xp 4859
This theorem is referenced by:  poinxp  4917  soinxp  4918  frinxp  4919  seinxp  4920  exfo  6055  isores2  6239  ltpiord  9319  ordpinq  9375  pwsleval  15390  tsrss  16468  ordtrest  20216  ordtrest2lem  20217  ordtrestNEW  28735  ordtrest2NEWlem  28736
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