Theorem cnvsymOLD 4305

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

Assertion

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

Distinct variable group:   x,y,R

Proof of Theorem cnvsymOLD

Step	Hyp	Ref	Expression
1		df-cnv 4002	. . . . 5 |- `'R = {<.y, x>. | xRy}
2	1	sseq1i 2641	. . . 4 |- (`'R C_ R <-> {<.y, x>. | xRy} C_ R)
3		ssel 2615	. . . . . 6 |- ({<.y, x>. | xRy} C_ R -> (<.y, x>. e. {<.y, x>. | xRy} -> <.y, x>. e. R))
4		df-br 3339	. . . . . 6 |- (yRx <-> <.y, x>. e. R)
5	3, 4	syl6ibr 230	. . . . 5 |- ({<.y, x>. | xRy} C_ R -> (<.y, x>. e. {<.y, x>. | xRy} -> yRx))
6		opabid 3557	. . . . 5 |- (<.y, x>. e. {<.y, x>. | xRy} <-> xRy)
7	5, 6	syl5ibr 224	. . . 4 |- ({<.y, x>. | xRy} C_ R -> (xRy -> yRx))
8	2, 7	sylbi 216	. . 3 |- (`'R C_ R -> (xRy -> yRx))
9	8	19.21aivv 1665	. 2 |- (`'R C_ R -> A.xA.y(xRy -> yRx))
10		ssopab2 3573	. . . . 5 |- ({<.y, x>. | xRy} C_ {<.y, x>. | yRx} <-> A.yA.x(xRy -> yRx))
11		alcom 1379	. . . . 5 |- (A.yA.x(xRy -> yRx) <-> A.xA.y(xRy -> yRx))
12	10, 11	bitri 190	. . . 4 |- ({<.y, x>. | xRy} C_ {<.y, x>. | yRx} <-> A.xA.y(xRy -> yRx))
13		opabss 3397	. . . . 5 |- {<.y, x>. | yRx} C_ R
14		sstr2 2623	. . . . 5 |- ({<.y, x>. | xRy} C_ {<.y, x>. | yRx} -> ({<.y, x>. | yRx} C_ R -> {<.y, x>. | xRy} C_ R))
15	13, 14	mpi 55	. . . 4 |- ({<.y, x>. | xRy} C_ {<.y, x>. | yRx} -> {<.y, x>. | xRy} C_ R)
16	12, 15	sylbir 218	. . 3 |- (A.xA.y(xRy -> yRx) -> {<.y, x>. | xRy} C_ R)
17	16, 1	syl5ss 2661	. 2 |- (A.xA.y(xRy -> yRx) -> `'R C_ R)
18	9, 17	impbii 174	1 |- (`'R C_ R <-> A.xA.y(xRy -> yRx))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   e. wcel 1300   C_ wss 2593  <.cop 3046   class class class wbr 3338  {copab 3395  `'ccnv 3985

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-cnv 4002

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