Theorem cnvsym 3429

Description: Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51.

Assertion

Ref	Expression
cnvsym	|- (`'R (_ R <-> A.xA.y(xRy -> yRx))

Distinct variable group:   x,y,R

Proof of Theorem cnvsym

Step	Hyp	Ref	Expression
1		df-cnv 3181	. . . . 5 |- `'R = {<.y, x>. | xRy}
2	1	sseq1i 2081	. . . 4 |- (`'R (_ R <-> {<.y, x>. | xRy} (_ R)
3		ssel 2059	. . . . . 6 |- ({<.y, x>. | xRy} (_ R -> (<.y, x>. e. {<.y, x>. | xRy} -> <.y, x>. e. R))
4		df-br 2615	. . . . . 6 |- (yRx <-> <.y, x>. e. R)
5	3, 4	syl6ibr 213	. . . . 5 |- ({<.y, x>. | xRy} (_ R -> (<.y, x>. e. {<.y, x>. | xRy} -> yRx))
6		opabid 2805	. . . . 5 |- (<.y, x>. e. {<.y, x>. | xRy} <-> xRy)
7	5, 6	syl5ibr 207	. . . 4 |- ({<.y, x>. | xRy} (_ R -> (xRy -> yRx))
8	2, 7	sylbi 199	. . 3 |- (`'R (_ R -> (xRy -> yRx))
9	8	19.21aivv 1285	. 2 |- (`'R (_ R -> A.xA.y(xRy -> yRx))
10		ssopab2 2817	. . . . 5 |- ({<.y, x>. | xRy} (_ {<.y, x>. | yRx} <-> A.yA.x(xRy -> yRx))
11		alcom 1030	. . . . 5 |- (A.yA.x(xRy -> yRx) <-> A.xA.y(xRy -> yRx))
12	10, 11	bitr 173	. . . 4 |- ({<.y, x>. | xRy} (_ {<.y, x>. | yRx} <-> A.xA.y(xRy -> yRx))
13		opabss 2663	. . . . 5 |- {<.y, x>. | yRx} (_ R
14		sstr2 2067	. . . . 5 |- ({<.y, x>. | xRy} (_ {<.y, x>. | yRx} -> ({<.y, x>. | yRx} (_ R -> {<.y, x>. | xRy} (_ R))
15	13, 14	mpi 44	. . . 4 |- ({<.y, x>. | xRy} (_ {<.y, x>. | yRx} -> {<.y, x>. | xRy} (_ R)
16	12, 15	sylbir 201	. . 3 |- (A.xA.y(xRy -> yRx) -> {<.y, x>. | xRy} (_ R)
17	16, 1	syl5ss 2101	. 2 |- (A.xA.y(xRy -> yRx) -> `'R (_ R)
18	9, 17	impbi 157	1 |- (`'R (_ R <-> A.xA.y(xRy -> yRx))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952   e. wcel 956   (_ wss 2043  <.cop 2407   class class class wbr 2614  {copab 2661  `'ccnv 3164

This theorem is referenced by:  dfer2 4252

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-cnv 3181

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