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Theorem relcnvexb 5349
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 5347 . 2  |-  ( R  e.  _V  ->  `' R  e.  _V )
2 dfrel2 5263 . . 3  |-  ( Rel 
R  <->  `' `' R  =  R
)
3 cnvexg 5347 . . . 4  |-  ( `' R  e.  _V  ->  `' `' R  e.  _V )
4 eleq1 2449 . . . 4  |-  ( `' `' R  =  R  ->  ( `' `' R  e.  _V  <->  R  e.  _V ) )
53, 4syl5ib 211 . . 3  |-  ( `' `' R  =  R  ->  ( `' R  e. 
_V  ->  R  e.  _V ) )
62, 5sylbi 188 . 2  |-  ( Rel 
R  ->  ( `' R  e.  _V  ->  R  e.  _V ) )
71, 6impbid2 196 1  |-  ( Rel 
R  ->  ( R  e.  _V  <->  `' R  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   _Vcvv 2901   `'ccnv 4819   Rel wrel 4825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-xp 4826  df-rel 4827  df-cnv 4828  df-dm 4830  df-rn 4831
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