Theorem ersym 6449

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 6446	. . . . . 6 |- ( R Er X -> Rel R )
4	2, 3	syl 14	. . . . 5 |- ( ph -> Rel R )
5		brrelex12 4585	. . . . 5 |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )
6	4, 1, 5	syl2anc 409	. . . 4 |- ( ph -> ( A e. _V /\ B e. _V ) )
7		brcnvg 4728	. . . . 5 |- ( ( B e. _V /\ A e. _V ) -> ( B `' R A <-> A R B ) )
8	7	ancoms 266	. . . 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 166	. 2 |- ( ph -> B `' R A )
11		df-er 6437	. . . . . 6 |- ( R Er X <-> ( Rel R /\ dom R = X /\ ( `' R u. ( R o. R ) ) C_ R ) )
12	11	simp3bi 999	. . . . 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 3258	. . 3 |- ( ph -> `' R C_ R )
15	14	ssbrd 3979	. 2 |- ( ph -> ( B `' R A -> B R A ) )
16	10, 15	mpd 13	1 |- ( ph -> B R A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   _Vcvv 2689   u. cun 3074   C_ wss 3076   class class class wbr 3937   `'ccnv 4546   dom cdm 4547   o. ccom 4551   Rel wrel 4552   Er wer 6434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-er 6437

This theorem is referenced by:  ercl2  6450  ersymb  6451  ertr2d  6454  ertr3d  6455  ertr4d  6456  erth  6481  erinxp  6511