Theorem ersym 6809

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 6806	. . . . . 6 |- ( R Er X -> Rel R )
4	2, 3	syl 14	. . . . 5 |- ( ph -> Rel R )
5		brrelex12 4808	. . . . 5 |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )
6	4, 1, 5	syl2anc 415	. . . 4 |- ( ph -> ( A e. _V /\ B e. _V ) )
7		brcnvg 4956	. . . . 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 6797	. . . . . 6 |- ( R Er X <-> ( Rel R /\ dom R = X /\ ( `' R u. ( R o. R ) ) C_ R ) )
12	11	simp3bi 1045	. . . . 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 3406	. . 3 |- ( ph -> `' R C_ R )
15	14	ssbrd 4168	. 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 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   C_ wss 3220   class class class wbr 4125   `'ccnv 4768   dom cdm 4769   o. ccom 4773   Rel wrel 4774   Er wer 6794

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-er 6797

This theorem is referenced by:  ercl2  6810  ersymb  6811  ertr2d  6814  ertr3d  6815  ertr4d  6816  erth  6843  erinxp  6873  qusgrp2  13893  2idlcpblrng  14832