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Theorem brres 5278
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 5277 . 2  |-  ( <. A ,  B >.  e.  ( C  |`  D )  <-> 
( <. A ,  B >.  e.  C  /\  A  e.  D ) )
3 df-br 4448 . 2  |-  ( A ( C  |`  D ) B  <->  <. A ,  B >.  e.  ( C  |`  D ) )
4 df-br 4448 . . 3  |-  ( A C B  <->  <. A ,  B >.  e.  C )
54anbi1i 695 . 2  |-  ( ( A C B  /\  A  e.  D )  <->  (
<. A ,  B >.  e.  C  /\  A  e.  D ) )
62, 3, 53bitr4i 277 1  |-  ( A ( C  |`  D ) B  <->  ( A C B  /\  A  e.  D ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1767   _Vcvv 3113   <.cop 4033   class class class wbr 4447    |` cres 5001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-res 5011
This theorem is referenced by:  dfres2  5324  dfima2  5337  poirr2  5389  cores  5508  resco  5509  rnco  5511  fnres  5695  fvres  5878  nfunsn  5895  1stconst  6868  2ndconst  6869  fsplit  6885  dprd2da  16881  metustidOLD  20797  metustid  20798  dvres  22050  dvres2  22051  axhcompl-zf  25591  hlimadd  25786  hhcmpl  25793  hhcms  25796  hlim0  25829  dfpo2  28761  dfdm5  28783  dfrn5  28784  wfrlem5  28924  frrlem5  28968  txpss3v  29105  brtxp  29107  pprodss4v  29111  brpprod  29112  brimg  29164  brapply  29165  funpartfun  29170  dfrdg4  29177  funressnfv  31680  dfdfat2  31683
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