Theorem relcnveq3 34092

Description: Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.)

Assertion

Ref	Expression
relcnveq3	|- ( Rel R -> ( R = `' R <-> A. x A. y ( x R y -> y R x ) ) )

Distinct variable group:   x, R, y

Proof of Theorem relcnveq3

Step	Hyp	Ref	Expression
1		eqss 3618	. 2 |- ( R = `' R <-> ( R C_ `' R /\ `' R C_ R ) )
2		cnvsym 5510	. . . . . . 7 |- ( `' R C_ R <-> A. x A. y ( x R y -> y R x ) )
3	2	biimpi 206	. . . . . 6 |- ( `' R C_ R -> A. x A. y ( x R y -> y R x ) )
4	3	a1d 25	. . . . 5 |- ( `' R C_ R -> ( Rel R -> A. x A. y ( x R y -> y R x ) ) )
5	4	adantl 482	. . . 4 |- ( ( R C_ `' R /\ `' R C_ R ) -> ( Rel R -> A. x A. y ( x R y -> y R x ) ) )
6	5	com12 32	. . 3 |- ( Rel R -> ( ( R C_ `' R /\ `' R C_ R ) -> A. x A. y ( x R y -> y R x ) ) )
7		dfrel2 5583	. . . . 5 |- ( Rel R <-> `' `' R = R )
8		cnvss 5294	. . . . . . . 8 |- ( `' R C_ R -> `' `' R C_ `' R )
9		sseq1 3626	. . . . . . . 8 |- ( `' `' R = R -> ( `' `' R C_ `' R <-> R C_ `' R ) )
10	8, 9	syl5ibcom 235	. . . . . . 7 |- ( `' R C_ R -> ( `' `' R = R -> R C_ `' R ) )
11	2, 10	sylbir 225	. . . . . 6 |- ( A. x A. y ( x R y -> y R x ) -> ( `' `' R = R -> R C_ `' R ) )
12	11	com12 32	. . . . 5 |- ( `' `' R = R -> ( A. x A. y ( x R y -> y R x ) -> R C_ `' R ) )
13	7, 12	sylbi 207	. . . 4 |- ( Rel R -> ( A. x A. y ( x R y -> y R x ) -> R C_ `' R ) )
14	2	biimpri 218	. . . 4 |- ( A. x A. y ( x R y -> y R x ) -> `' R C_ R )
15	13, 14	jca2 556	. . 3 |- ( Rel R -> ( A. x A. y ( x R y -> y R x ) -> ( R C_ `' R /\ `' R C_ R ) ) )
16	6, 15	impbid 202	. 2 |- ( Rel R -> ( ( R C_ `' R /\ `' R C_ R ) <-> A. x A. y ( x R y -> y R x ) ) )
17	1, 16	syl5bb 272	1 |- ( Rel R -> ( R = `' R <-> A. x A. y ( x R y -> y R x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   C_ wss 3574   class class class wbr 4653   `'ccnv 5113   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by:  relcnveq  34093