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Theorem relcnvexb 6732
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 6730 . 2  |-  ( R  e.  _V  ->  `' R  e.  _V )
2 dfrel2 5274 . . 3  |-  ( Rel 
R  <->  `' `' R  =  R
)
3 cnvexg 6730 . . . 4  |-  ( `' R  e.  _V  ->  `' `' R  e.  _V )
4 eleq1 2474 . . . 4  |-  ( `' `' R  =  R  ->  ( `' `' R  e.  _V  <->  R  e.  _V ) )
53, 4syl5ib 219 . . 3  |-  ( `' `' R  =  R  ->  ( `' R  e. 
_V  ->  R  e.  _V ) )
62, 5sylbi 195 . 2  |-  ( Rel 
R  ->  ( `' R  e.  _V  ->  R  e.  _V ) )
71, 6impbid2 204 1  |-  ( Rel 
R  ->  ( R  e.  _V  <->  `' R  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1405    e. wcel 1842   _Vcvv 3059   `'ccnv 4822   Rel wrel 4828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-cnv 4831  df-dm 4833  df-rn 4834
This theorem is referenced by:  f1oexrnex  6733
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