Theorem ersym 6601

Description: An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
ersym.1	|- ( ph -> R Er X )
ersym.2	|- ( ph -> A R B )

Assertion

Ref	Expression
ersym	|- ( ph -> B R A )

Proof of Theorem ersym

Step	Hyp	Ref	Expression
1		ersym.2	. . 3 |- ( ph -> A R B )
2		ersym.1	. . . . . 6 |- ( ph -> R Er X )
3		errel 6598	. . . . . 6 |- ( R Er X -> Rel R )
4	2, 3	syl 14	. . . . 5 |- ( ph -> Rel R )
5		brrelex12 4698	. . . . 5 |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )
6	4, 1, 5	syl2anc 411	. . . 4 |- ( ph -> ( A e. _V /\ B e. _V ) )
7		brcnvg 4844	. . . . 5 |- ( ( B e. _V /\ A e. _V ) -> ( B `' R A <-> A R B ) )
8	7	ancoms 268	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( B `' R A <-> A R B ) )
9	6, 8	syl 14	. . 3 |- ( ph -> ( B `' R A <-> A R B ) )
10	1, 9	mpbird 167	. 2 |- ( ph -> B `' R A )
11		df-er 6589	. . . . . 6 |- ( R Er X <-> ( Rel R /\ dom R = X /\ ( `' R u. ( R o. R ) ) C_ R ) )
12	11	simp3bi 1016	. . . . 5 |- ( R Er X -> ( `' R u. ( R o. R ) ) C_ R )
13	2, 12	syl 14	. . . 4 |- ( ph -> ( `' R u. ( R o. R ) ) C_ R )
14	13	unssad 3337	. . 3 |- ( ph -> `' R C_ R )
15	14	ssbrd 4073	. 2 |- ( ph -> ( B `' R A -> B R A ) )
16	10, 15	mpd 13	1 |- ( ph -> B R A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   u. cun 3152   C_ wss 3154   class class class wbr 4030   `'ccnv 4659   dom cdm 4660   o. ccom 4664   Rel wrel 4665   Er wer 6586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-er 6589

This theorem is referenced by:  ercl2  6602  ersymb  6603  ertr2d  6606  ertr3d  6607  ertr4d  6608  erth  6635  erinxp  6665  qusgrp2  13186  2idlcpblrng  14022