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Theorem brinxp2 4881
Description: Intersection of binary relation with cross product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
brinxp2  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A  e.  C  /\  B  e.  D  /\  A R B ) )

Proof of Theorem brinxp2
StepHypRef Expression
1 brin 4202 . 2  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A R B  /\  A ( C  X.  D ) B ) )
2 ancom 438 . 2  |-  ( ( A R B  /\  A ( C  X.  D ) B )  <-> 
( A ( C  X.  D ) B  /\  A R B ) )
3 brxp 4851 . . . 4  |-  ( A ( C  X.  D
) B  <->  ( A  e.  C  /\  B  e.  D ) )
43anbi1i 677 . . 3  |-  ( ( A ( C  X.  D ) B  /\  A R B )  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
5 df-3an 938 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
64, 5bitr4i 244 . 2  |-  ( ( A ( C  X.  D ) B  /\  A R B )  <->  ( A  e.  C  /\  B  e.  D  /\  A R B ) )
71, 2, 63bitri 263 1  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A  e.  C  /\  B  e.  D  /\  A R B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1717    i^i cin 3264   class class class wbr 4155    X. cxp 4818
This theorem is referenced by:  brinxp  4882  fncnv  5457  erinxp  6916  fpwwe2lem8  8447  fpwwe2lem9  8448  fpwwe2lem12  8451  nqerf  8742  nqerid  8745  isstruct  13408  pwsle  13643  psss  14575  psssdm2  14576  pi1cpbl  18942  pi1grplem  18947
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-xp 4826
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