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Theorem relcnvexb 5183
Description: A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
Assertion
Ref Expression
relcnvexb  |-  ( Rel 
R  ->  ( R  e.  _V  <->  `' R  e.  _V ) )

Proof of Theorem relcnvexb
StepHypRef Expression
1 cnvexg 5181 . 2  |-  ( R  e.  _V  ->  `' R  e.  _V )
2 dfrel2 5094 . . 3  |-  ( Rel 
R  <->  `' `' R  =  R
)
3 cnvexg 5181 . . . 4  |-  ( `' R  e.  _V  ->  `' `' R  e.  _V )
4 eleq1 2252 . . . 4  |-  ( `' `' R  =  R  ->  ( `' `' R  e.  _V  <->  R  e.  _V ) )
53, 4imbitrid 154 . . 3  |-  ( `' `' R  =  R  ->  ( `' R  e. 
_V  ->  R  e.  _V ) )
62, 5sylbi 121 . 2  |-  ( Rel 
R  ->  ( `' R  e.  _V  ->  R  e.  _V ) )
71, 6impbid2 143 1  |-  ( Rel 
R  ->  ( R  e.  _V  <->  `' R  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1364    e. wcel 2160   _Vcvv 2752   `'ccnv 4640   Rel wrel 4646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4647  df-rel 4648  df-cnv 4649  df-dm 4651  df-rn 4652
This theorem is referenced by: (None)
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