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Theorem relcnvexb 5399
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 5397 . 2  |-  ( R  e.  _V  ->  `' R  e.  _V )
2 dfrel2 5313 . . 3  |-  ( Rel 
R  <->  `' `' R  =  R
)
3 cnvexg 5397 . . . 4  |-  ( `' R  e.  _V  ->  `' `' R  e.  _V )
4 eleq1 2495 . . . 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 1652    e. wcel 1725   _Vcvv 2948   `'ccnv 4869   Rel wrel 4875
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-13 1727  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-pow 4369  ax-pr 4395  ax-un 4693
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-eu 2284  df-mo 2285  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-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881
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