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Theorem brresg 5146
Description: Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
Assertion
Ref Expression
brresg  |-  ( B  e.  V  ->  ( A ( C  |`  D ) B  <->  ( A C B  /\  A  e.  D ) ) )

Proof of Theorem brresg
StepHypRef Expression
1 opelresg 5145 . 2  |-  ( B  e.  V  ->  ( <. A ,  B >.  e.  ( C  |`  D )  <-> 
( <. A ,  B >.  e.  C  /\  A  e.  D ) ) )
2 df-br 4205 . 2  |-  ( A ( C  |`  D ) B  <->  <. A ,  B >.  e.  ( C  |`  D ) )
3 df-br 4205 . . 3  |-  ( A C B  <->  <. A ,  B >.  e.  C )
43anbi1i 677 . 2  |-  ( ( A C B  /\  A  e.  D )  <->  (
<. A ,  B >.  e.  C  /\  A  e.  D ) )
51, 2, 43bitr4g 280 1  |-  ( B  e.  V  ->  ( A ( C  |`  D ) B  <->  ( A C B  /\  A  e.  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725   <.cop 3809   class class class wbr 4204    |` cres 4872
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-res 4882
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