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Theorem brin 4219
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 3490 . 2  |-  ( <. A ,  B >.  e.  ( R  i^i  S
)  <->  ( <. A ,  B >.  e.  R  /\  <. A ,  B >.  e.  S ) )
2 df-br 4173 . 2  |-  ( A ( R  i^i  S
) B  <->  <. A ,  B >.  e.  ( R  i^i  S ) )
3 df-br 4173 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4173 . . 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 1721    i^i cin 3279   <.cop 3777   class class class wbr 4172
This theorem is referenced by:  brinxp2  4898  trin2  5216  poirr2  5217  tpostpos  6458  erinxp  6937  sbthcl  7188  infxpenlem  7851  fpwwe2lem12  8472  fpwwe2  8474  isinv  13940  isffth2  14068  ffthf1o  14071  ffthoppc  14076  ffthres2c  14092  isunit  15717  opsrtoslem2  16500  dfpo2  25326  brtxp  25634  idsset  25644  dfon3  25646  elfix  25657  dffix2  25659  brcap  25693  funpartlem  25695  trer  26209  fneval  26257  fnessref  26263  refssfne  26264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-in 3287  df-br 4173
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