Theorem twsymr 14394

Description: Two ways of saying a relation is symmetric.

Assertion

Ref	Expression
twsymr	|- (Rel R -> (R = `'R <-> A.xA.y(xRy -> yRx)))

Distinct variable group:   x,R,y

Proof of Theorem twsymr

Step	Hyp	Ref	Expression
1		cnvsym 4304	. . . . . . 7 |- (`'R C_ R <-> A.xA.y(xRy -> yRx))
2	1	biimpi 168	. . . . . 6 |- (`'R C_ R -> A.xA.y(xRy -> yRx))
3	2	a1d 15	. . . . 5 |- (`'R C_ R -> (Rel R -> A.xA.y(xRy -> yRx)))
4	3	adantl 424	. . . 4 |- ((R C_ `'R /\ `'R C_ R) -> (Rel R -> A.xA.y(xRy -> yRx)))
5	4	com12 14	. . 3 |- (Rel R -> ((R C_ `'R /\ `'R C_ R) -> A.xA.y(xRy -> yRx)))
6		dfrel2 4358	. . . . . . 7 |- (Rel R <-> `'`'R = R)
7		sseq1 2637	. . . . . . . . . 10 |- (`'`'R = R -> (`'`'R C_ `'R <-> R C_ `'R))
8		cnvss 4134	. . . . . . . . . 10 |- (`'R C_ R -> `'`'R C_ `'R)
9	7, 8	syl5cbi 226	. . . . . . . . 9 |- (`'R C_ R -> (`'`'R = R -> R C_ `'R))
10	1, 9	sylbir 218	. . . . . . . 8 |- (A.xA.y(xRy -> yRx) -> (`'`'R = R -> R C_ `'R))
11	10	com12 14	. . . . . . 7 |- (`'`'R = R -> (A.xA.y(xRy -> yRx) -> R C_ `'R))
12	6, 11	sylbi 216	. . . . . 6 |- (Rel R -> (A.xA.y(xRy -> yRx) -> R C_ `'R))
13	12	imp 377	. . . . 5 |- ((Rel R /\ A.xA.y(xRy -> yRx)) -> R C_ `'R)
14	1	biimpri 169	. . . . . 6 |- (A.xA.y(xRy -> yRx) -> `'R C_ R)
15	14	adantl 424	. . . . 5 |- ((Rel R /\ A.xA.y(xRy -> yRx)) -> `'R C_ R)
16	13, 15	jca 310	. . . 4 |- ((Rel R /\ A.xA.y(xRy -> yRx)) -> (R C_ `'R /\ `'R C_ R))
17	16	ex 402	. . 3 |- (Rel R -> (A.xA.y(xRy -> yRx) -> (R C_ `'R /\ `'R C_ R)))
18	5, 17	impbid 574	. 2 |- (Rel R -> ((R C_ `'R /\ `'R C_ R) <-> A.xA.y(xRy -> yRx)))
19		eqss 2631	. 2 |- (R = `'R <-> (R C_ `'R /\ `'R C_ R))
20	18, 19	syl5bb 591	1 |- (Rel R -> (R = `'R <-> A.xA.y(xRy -> yRx)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   C_ wss 2593   class class class wbr 3338  `'ccnv 3985  Rel wrel 3991

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-xp 4000  df-rel 4001  df-cnv 4002

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