MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brres Unicode version

Theorem brres 5064
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 5063 . 2  |-  ( <. A ,  B >.  e.  ( C  |`  D )  <-> 
( <. A ,  B >.  e.  C  /\  A  e.  D ) )
3 df-br 4126 . 2  |-  ( A ( C  |`  D ) B  <->  <. A ,  B >.  e.  ( C  |`  D ) )
4 df-br 4126 . . 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 1715   _Vcvv 2873   <.cop 3732   class class class wbr 4125    |` cres 4794
This theorem is referenced by:  dfres2  5105  dfima2  5117  poirr2  5170  cores  5279  resco  5280  rnco  5282  fnres  5465  fvres  5649  nfunsn  5665  1stconst  6335  2ndconst  6336  fsplit  6351  dprd2da  15487  dvres  19476  dvres2  19477  axhcompl-zf  21891  hlimadd  22085  hhcmpl  22092  hhcms  22095  hlim0  22128  metustid  23797  dfpo2  24853  dfdm5  24873  dfrn5  24874  wfrlem5  25001  frrlem5  25026  txpss3v  25159  brtxp  25161  pprodss4v  25165  brpprod  25166  brimg  25217  brapply  25218  funpartfun  25223  dfrdg4  25230  funressnfv  27499  dfdfat2  27502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-xp 4798  df-res 4804
  Copyright terms: Public domain W3C validator