Theorem ersym 6655

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 6652	. . . . . 6 |- ( R Er X -> Rel R )
4	2, 3	syl 14	. . . . 5 |- ( ph -> Rel R )
5		brrelex12 4731	. . . . 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 4877	. . . . 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 6643	. . . . . 6 |- ( R Er X <-> ( Rel R /\ dom R = X /\ ( `' R u. ( R o. R ) ) C_ R ) )
12	11	simp3bi 1017	. . . . 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 3358	. . 3 |- ( ph -> `' R C_ R )
15	14	ssbrd 4102	. 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 1373   e. wcel 2178   _Vcvv 2776   u. cun 3172   C_ wss 3174   class class class wbr 4059   `'ccnv 4692   dom cdm 4693   o. ccom 4697   Rel wrel 4698   Er wer 6640

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-er 6643

This theorem is referenced by:  ercl2  6656  ersymb  6657  ertr2d  6660  ertr3d  6661  ertr4d  6662  erth  6689  erinxp  6719  qusgrp2  13564  2idlcpblrng  14400