Theorem relcnvexb 5268

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 5266	. 2 |- ( R e. _V -> `' R e. _V )
2		dfrel2 5179	. . 3 |- ( Rel R <-> `' `' R = R )
3		cnvexg 5266	. . . 4 |- ( `' R e. _V -> `' `' R e. _V )
4		eleq1 2292	. . . 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 1395   e. wcel 2200   _Vcvv 2799   `'ccnv 4718   Rel wrel 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by: (None)