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Theorem brres 4961
Description: Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
Hypothesis
Ref Expression
opelres.1  |-  B  e. 
_V
Assertion
Ref Expression
brres  |-  ( A ( C  |`  D ) B  <->  ( A C B  /\  A  e.  D ) )

Proof of Theorem brres
StepHypRef Expression
1 opelres.1 . . 3  |-  B  e. 
_V
21opelres 4960 . 2  |-  ( <. A ,  B >.  e.  ( C  |`  D )  <-> 
( <. A ,  B >.  e.  C  /\  A  e.  D ) )
3 df-br 4024 . 2  |-  ( A ( C  |`  D ) B  <->  <. A ,  B >.  e.  ( C  |`  D ) )
4 df-br 4024 . . 3  |-  ( A C B  <->  <. A ,  B >.  e.  C )
54anbi1i 676 . 2  |-  ( ( A C B  /\  A  e.  D )  <->  (
<. A ,  B >.  e.  C  /\  A  e.  D ) )
62, 3, 53bitr4i 268 1  |-  ( A ( C  |`  D ) B  <->  ( A C B  /\  A  e.  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023    |` cres 4691
This theorem is referenced by:  dfres2  5002  dfima2  5014  poirr2  5067  cores  5176  resco  5177  rnco  5179  fnres  5360  fvres  5542  nfunsn  5558  1stconst  6207  2ndconst  6208  fsplit  6223  dprd2da  15277  dvres  19261  dvres2  19262  axhcompl-zf  21578  hlimadd  21772  hhcmpl  21779  hhcms  21782  hlim0  21815  dfpo2  24112  dfdm5  24132  dfrn5  24133  wfrlem5  24260  frrlem5  24285  txpss3v  24418  brtxp  24420  pprodss4v  24424  brpprod  24425  brimg  24476  brapply  24477  funpartfun  24481  dfrdg4  24488  funressnfv  27991  dfdfat2  27994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-res 4701
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