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Theorem brres 4793
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 4792 . 2  |-  ( <. A ,  B >.  e.  ( C  |`  D )  <-> 
( <. A ,  B >.  e.  C  /\  A  e.  D ) )
3 df-br 3898 . 2  |-  ( A ( C  |`  D ) B  <->  <. A ,  B >.  e.  ( C  |`  D ) )
4 df-br 3898 . . 3  |-  ( A C B  <->  <. A ,  B >.  e.  C )
54anbi1i 451 . 2  |-  ( ( A C B  /\  A  e.  D )  <->  (
<. A ,  B >.  e.  C  /\  A  e.  D ) )
62, 3, 53bitr4i 211 1  |-  ( A ( C  |`  D ) B  <->  ( A C B  /\  A  e.  D ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 1463   _Vcvv 2658   <.cop 3498   class class class wbr 3897    |` cres 4509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-res 4519
This theorem is referenced by:  dfres2  4839  dfima2  4851  poirr2  4899  cores  5010  resco  5011  rnco  5013  fnres  5207  fvres  5411  nfunsn  5421  1stconst  6084  2ndconst  6085
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