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Theorem brin 3858
Description: The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
Assertion
Ref Expression
brin  |-  ( A ( R  i^i  S
) B  <->  ( A R B  /\  A S B ) )

Proof of Theorem brin
StepHypRef Expression
1 elin 3167 . 2  |-  ( <. A ,  B >.  e.  ( R  i^i  S
)  <->  ( <. A ,  B >.  e.  R  /\  <. A ,  B >.  e.  S ) )
2 df-br 3812 . 2  |-  ( A ( R  i^i  S
) B  <->  <. A ,  B >.  e.  ( R  i^i  S ) )
3 df-br 3812 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 3812 . . 3  |-  ( A S B  <->  <. A ,  B >.  e.  S )
53, 4anbi12i 448 . 2  |-  ( ( A R B  /\  A S B )  <->  ( <. A ,  B >.  e.  R  /\  <. A ,  B >.  e.  S ) )
61, 2, 53bitr4i 210 1  |-  ( A ( R  i^i  S
) B  <->  ( A R B  /\  A S B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    e. wcel 1434    i^i cin 2983   <.cop 3425   class class class wbr 3811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990  df-br 3812
This theorem is referenced by:  brinxp2  4463  trin2  4778  poirr2  4779  cnvin  4793  tpostpos  5961  erinxp  6296
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