Theorem ercl 6691

Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Hypotheses

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

Assertion

Ref	Expression
ercl	|- ( ph -> A e. X )

Proof of Theorem ercl

Step	Hyp	Ref	Expression
1		ersym.1	. . . 4 |- ( ph -> R Er X )
2		errel 6689	. . . 4 |- ( R Er X -> Rel R )
3	1, 2	syl 14	. . 3 |- ( ph -> Rel R )
4		ersym.2	. . 3 |- ( ph -> A R B )
5		releldm 4959	. . 3 |- ( ( Rel R /\ A R B ) -> A e. dom R )
6	3, 4, 5	syl2anc 411	. 2 |- ( ph -> A e. dom R )
7		erdm 6690	. . 3 |- ( R Er X -> dom R = X )
8	1, 7	syl 14	. 2 |- ( ph -> dom R = X )
9	6, 8	eleqtrd 2308	1 |- ( ph -> A e. X )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   class class class wbr 4083   dom cdm 4719   Rel wrel 4724   Er wer 6677

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-dm 4729  df-er 6680

This theorem is referenced by:  ercl2  6693  erthi  6728  qliftfun  6764  qusgrp2  13650