Theorem cnvsym 3443

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 3192	. . . . 5 |- `'R = {<.y, x>. | xRy}
2	1	sseq1i 2088	. . . 4 |- (`'R (_ R <-> {<.y, x>. | xRy} (_ R)
3		ssel 2066	. . . . . 6 |- ({<.y, x>. | xRy} (_ R -> (<.y, x>. e. {<.y, x>. | xRy} -> <.y, x>. e. R))
4		df-br 2625	. . . . . 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 2816	. . . . 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 1289	. 2 |- (`'R (_ R -> A.xA.y(xRy -> yRx))
10		ssopab2 2828	. . . . 5 |- ({<.y, x>. | xRy} (_ {<.y, x>. | yRx} <-> A.yA.x(xRy -> yRx))
11		alcom 1034	. . . . 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 2673	. . . . 5 |- {<.y, x>. | yRx} (_ R
14		sstr2 2074	. . . . 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 2108	. 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 956   e. wcel 960   (_ wss 2050  <.cop 2415   class class class wbr 2624  {copab 2671  `'ccnv 3175

This theorem is referenced by:  dfer2 4268

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-cnv 3192

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