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Theorem relcnvexb 6724
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 6722 . 2  |-  ( R  e.  _V  ->  `' R  e.  _V )
2 dfrel2 5450 . . 3  |-  ( Rel 
R  <->  `' `' R  =  R
)
3 cnvexg 6722 . . . 4  |-  ( `' R  e.  _V  ->  `' `' R  e.  _V )
4 eleq1 2534 . . . 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 1374    e. wcel 1762   _Vcvv 3108   `'ccnv 4993   Rel wrel 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-cnv 5002  df-dm 5004  df-rn 5005
This theorem is referenced by:  f1oexrnex  6725
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