Theorem ersym 7754

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 7751	. . . . . 6 |- ( R Er X -> Rel R )
4	2, 3	syl 17	. . . . 5 |- ( ph -> Rel R )
5		brrelex12 5155	. . . . 5 |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )
6	4, 1, 5	syl2anc 693	. . . 4 |- ( ph -> ( A e. _V /\ B e. _V ) )
7		brcnvg 5303	. . . . 5 |- ( ( B e. _V /\ A e. _V ) -> ( B `' R A <-> A R B ) )
8	7	ancoms 469	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( B `' R A <-> A R B ) )
9	6, 8	syl 17	. . 3 |- ( ph -> ( B `' R A <-> A R B ) )
10	1, 9	mpbird 247	. 2 |- ( ph -> B `' R A )
11		df-er 7742	. . . . . 6 |- ( R Er X <-> ( Rel R /\ dom R = X /\ ( `' R u. ( R o. R ) ) C_ R ) )
12	11	simp3bi 1078	. . . . 5 |- ( R Er X -> ( `' R u. ( R o. R ) ) C_ R )
13	2, 12	syl 17	. . . 4 |- ( ph -> ( `' R u. ( R o. R ) ) C_ R )
14	13	unssad 3790	. . 3 |- ( ph -> `' R C_ R )
15	14	ssbrd 4696	. 2 |- ( ph -> ( B `' R A -> B R A ) )
16	10, 15	mpd 15	1 |- ( ph -> B R A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   o. ccom 5118   Rel wrel 5119   Er wer 7739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-er 7742

This theorem is referenced by:  ercl2  7755  ersymb  7756  ertr2d  7759  ertr3d  7760  ertr4d  7761  erth  7791  erinxp  7821  nqereu  9751  nqerf  9752  1nqenq  9784  qusgrp2  17533  efginvrel2  18140  efgcpbllemb  18168  2idlcpbl  19234  tgptsmscls  21953