Theorem relcnvexb 3521

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 3519	. 2 |- (R e. V -> `'R e. V)
2		dfrel2 3485	. . 3 |- (Rel R <-> `'`'R = R)
3		eleq1 1534	. . . 4 |- (`'`'R = R -> (`'`'R e. V <-> R e. V))
4		cnvexg 3519	. . . 4 |- (`'R e. V -> `'`'R e. V)
5	3, 4	syl5bi 208	. . 3 |- (`'`'R = R -> (`'R e. V -> R e. V))
6	2, 5	sylbi 199	. 2 |- (Rel R -> (`'R e. V -> R e. V))
7	1, 6	impbid2 518	1 |- (Rel R -> (R e. V <-> `'R e. V))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  Vcvv 1811  `'ccnv 3169  Rel wrel 3175

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189

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