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Theorem brinxp2 5060
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 4496 . 2  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A R B  /\  A ( C  X.  D ) B ) )
2 ancom 450 . 2  |-  ( ( A R B  /\  A ( C  X.  D ) B )  <-> 
( A ( C  X.  D ) B  /\  A R B ) )
3 brxp 5029 . . . 4  |-  ( A ( C  X.  D
) B  <->  ( A  e.  C  /\  B  e.  D ) )
43anbi1i 695 . . 3  |-  ( ( A ( C  X.  D ) B  /\  A R B )  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
5 df-3an 975 . . 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 973    e. wcel 1767    i^i cin 3475   class class class wbr 4447    X. cxp 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005
This theorem is referenced by:  brinxp  5061  fncnv  5650  erinxp  7382  fpwwe2lem8  9011  fpwwe2lem9  9012  fpwwe2lem12  9015  nqerf  9304  nqerid  9307  isstruct  14496  pwsle  14743  psss  15697  psssdm2  15698  pi1cpbl  21279  pi1grplem  21284
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