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Theorem brin 4259
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 3530 . 2  |-  ( <. A ,  B >.  e.  ( R  i^i  S
)  <->  ( <. A ,  B >.  e.  R  /\  <. A ,  B >.  e.  S ) )
2 df-br 4213 . 2  |-  ( A ( R  i^i  S
) B  <->  <. A ,  B >.  e.  ( R  i^i  S ) )
3 df-br 4213 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4213 . . 3  |-  ( A S B  <->  <. A ,  B >.  e.  S )
53, 4anbi12i 679 . 2  |-  ( ( A R B  /\  A S B )  <->  ( <. A ,  B >.  e.  R  /\  <. A ,  B >.  e.  S ) )
61, 2, 53bitr4i 269 1  |-  ( A ( R  i^i  S
) B  <->  ( A R B  /\  A S B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1725    i^i cin 3319   <.cop 3817   class class class wbr 4212
This theorem is referenced by:  brinxp2  4939  trin2  5257  poirr2  5258  tpostpos  6499  erinxp  6978  sbthcl  7229  infxpenlem  7895  fpwwe2lem12  8516  fpwwe2  8518  isinv  13985  isffth2  14113  ffthf1o  14116  ffthoppc  14121  ffthres2c  14137  isunit  15762  opsrtoslem2  16545  dfpo2  25378  brtxp  25725  idsset  25735  dfon3  25737  elfix  25748  dffix2  25750  brcap  25785  funpartlem  25787  trer  26319  fneval  26367  fnessref  26373  refssfne  26374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-in 3327  df-br 4213
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