Theorem relcnvexb 5304

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

Step	Hyp	Ref	Expression
1		cnvexg 5302	. 2 |- ( R e. _V -> `' R e. _V )
2		dfrel2 5215	. . 3 |- ( Rel R <-> `' `' R = R )
3		cnvexg 5302	. . . 4 |- ( `' R e. _V -> `' `' R e. _V )
4		eleq1 2297	. . . 4 |- ( `' `' R = R -> ( `' `' R e. _V <-> R e. _V ) )
5	3, 4	imbitrid 154	. . 3 |- ( `' `' R = R -> ( `' R e. _V -> R e. _V ) )
6	2, 5	sylbi 121	. 2 |- ( Rel R -> ( `' R e. _V -> R e. _V ) )
7	1, 6	impbid2 143	1 |- ( Rel R -> ( R e. _V <-> `' R e. _V ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   `'ccnv 4750   Rel wrel 4756

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758  df-cnv 4759  df-dm 4761  df-rn 4762

This theorem is referenced by: (None)