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Theorem brinxp2 4896
Description: Intersection of binary relation with Cartesian 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 4338 . 2  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A R B  /\  A ( C  X.  D ) B ) )
2 ancom 448 . 2  |-  ( ( A R B  /\  A ( C  X.  D ) B )  <-> 
( A ( C  X.  D ) B  /\  A R B ) )
3 brxp 4866 . . . 4  |-  ( A ( C  X.  D
) B  <->  ( A  e.  C  /\  B  e.  D ) )
43anbi1i 690 . . 3  |-  ( ( A ( C  X.  D ) B  /\  A R B )  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
5 df-3an 962 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
64, 5bitr4i 252 . 2  |-  ( ( A ( C  X.  D ) B  /\  A R B )  <->  ( A  e.  C  /\  B  e.  D  /\  A R B ) )
71, 2, 63bitri 271 1  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A  e.  C  /\  B  e.  D  /\  A R B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 960    e. wcel 1761    i^i cin 3324   class class class wbr 4289    X. cxp 4834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-xp 4842
This theorem is referenced by:  brinxp  4897  fncnv  5479  erinxp  7170  fpwwe2lem8  8800  fpwwe2lem9  8801  fpwwe2lem12  8804  nqerf  9095  nqerid  9098  isstruct  14180  pwsle  14426  psss  15380  psssdm2  15381  pi1cpbl  20575  pi1grplem  20580
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