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Theorem brin 4411
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 3587 . 2  |-  ( <. A ,  B >.  e.  ( R  i^i  S
)  <->  ( <. A ,  B >.  e.  R  /\  <. A ,  B >.  e.  S ) )
2 df-br 4362 . 2  |-  ( A ( R  i^i  S
) B  <->  <. A ,  B >.  e.  ( R  i^i  S ) )
3 df-br 4362 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4362 . . 3  |-  ( A S B  <->  <. A ,  B >.  e.  S )
53, 4anbi12i 701 . 2  |-  ( ( A R B  /\  A S B )  <->  ( <. A ,  B >.  e.  R  /\  <. A ,  B >.  e.  S ) )
61, 2, 53bitr4i 280 1  |-  ( A ( R  i^i  S
) B  <->  ( A R B  /\  A S B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    e. wcel 1872    i^i cin 3373   <.cop 3942   class class class wbr 4361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-v 3019  df-in 3381  df-br 4362
This theorem is referenced by:  brinxp2  4853  trin2  5180  poirr2  5181  tpostpos  6943  erinxp  7387  sbthcl  7642  infxpenlem  8391  fpwwe2lem12  9012  fpwwe2  9014  isinv  15603  isffth2  15759  ffthf1o  15762  ffthoppc  15767  ffthres2c  15783  isunit  17823  opsrtoslem2  18646  posrasymb  28364  trleile  28373  dfpo2  30341  brtxp  30591  idsset  30601  dfon3  30603  elfix  30614  dffix2  30616  brcap  30651  funpartlem  30653  trer  30916  fneval  30952
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