Theorem ercl 7753

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 7751	. . . 4 |- ( R Er X -> Rel R )
3	1, 2	syl 17	. . 3 |- ( ph -> Rel R )
4		ersym.2	. . 3 |- ( ph -> A R B )
5		releldm 5358	. . 3 |- ( ( Rel R /\ A R B ) -> A e. dom R )
6	3, 4, 5	syl2anc 693	. 2 |- ( ph -> A e. dom R )
7		erdm 7752	. . 3 |- ( R Er X -> dom R = X )
8	1, 7	syl 17	. 2 |- ( ph -> dom R = X )
9	6, 8	eleqtrd 2703	1 |- ( ph -> A e. X )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   class class class wbr 4653   dom cdm 5114   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-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-dm 5124  df-er 7742

This theorem is referenced by:  ercl2  7755  erthi  7793  qliftfun  7832  efgcpbl2  18170  frgpcpbl  18172  prter3  34167