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Theorem brresg 4668
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 4667 . 2  |-  ( B  e.  V  ->  ( <. A ,  B >.  e.  ( C  |`  D )  <-> 
( <. A ,  B >.  e.  C  /\  A  e.  D ) ) )
2 df-br 3806 . 2  |-  ( A ( C  |`  D ) B  <->  <. A ,  B >.  e.  ( C  |`  D ) )
3 df-br 3806 . . 3  |-  ( A C B  <->  <. A ,  B >.  e.  C )
43anbi1i 446 . 2  |-  ( ( A C B  /\  A  e.  D )  <->  (
<. A ,  B >.  e.  C  /\  A  e.  D ) )
51, 2, 43bitr4g 221 1  |-  ( B  e.  V  ->  ( A ( C  |`  D ) B  <->  ( A C B  /\  A  e.  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1434   <.cop 3419   class class class wbr 3805    |` cres 4393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-res 4403
This theorem is referenced by: (None)
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