Theorem ercl 6712

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 6710	. . . 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 4967	. . 3 |- ( ( Rel R /\ A R B ) -> A e. dom R )
6	3, 4, 5	syl2anc 411	. 2 |- ( ph -> A e. dom R )
7		erdm 6711	. . 3 |- ( R Er X -> dom R = X )
8	1, 7	syl 14	. 2 |- ( ph -> dom R = X )
9	6, 8	eleqtrd 2310	1 |- ( ph -> A e. X )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   class class class wbr 4088   dom cdm 4725   Rel wrel 4730   Er wer 6698

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-dm 4735  df-er 6701

This theorem is referenced by:  ercl2  6714  erthi  6749  qliftfun  6785  qusgrp2  13699