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Theorem brinxp2 4678
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 4041 . 2  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A R B  /\  A ( C  X.  D ) B ) )
2 ancom 264 . 2  |-  ( ( A R B  /\  A ( C  X.  D ) B )  <-> 
( A ( C  X.  D ) B  /\  A R B ) )
3 brxp 4642 . . . 4  |-  ( A ( C  X.  D
) B  <->  ( A  e.  C  /\  B  e.  D ) )
43anbi1i 455 . . 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 186 . 2  |-  ( ( A ( C  X.  D ) B  /\  A R B )  <->  ( A  e.  C  /\  B  e.  D  /\  A R B ) )
71, 2, 63bitri 205 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:    /\ wa 103    <-> wb 104    /\ w3a 973    e. wcel 2141    i^i cin 3120   class class class wbr 3989    X. cxp 4609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617
This theorem is referenced by:  brinxp  4679  fncnv  5264  erinxp  6587  isstructim  12430  isstructr  12431
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