Theorem erref 7762

Description: An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
ersymb.1	|- ( ph -> R Er X )
erref.2	|- ( ph -> A e. X )

Assertion

Ref	Expression
erref	|- ( ph -> A R A )

Proof of Theorem erref

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		erref.2	. . . 4 |- ( ph -> A e. X )
2		ersymb.1	. . . . 5 |- ( ph -> R Er X )
3		erdm 7752	. . . . 5 |- ( R Er X -> dom R = X )
4	2, 3	syl 17	. . . 4 |- ( ph -> dom R = X )
5	1, 4	eleqtrrd 2704	. . 3 |- ( ph -> A e. dom R )
6		eldmg 5319	. . . 4 |- ( A e. X -> ( A e. dom R <-> E. x A R x ) )
7	1, 6	syl 17	. . 3 |- ( ph -> ( A e. dom R <-> E. x A R x ) )
8	5, 7	mpbid 222	. 2 |- ( ph -> E. x A R x )
9	2	adantr 481	. . 3 |- ( ( ph /\ A R x ) -> R Er X )
10		simpr 477	. . 3 |- ( ( ph /\ A R x ) -> A R x )
11	9, 10, 10	ertr4d 7761	. 2 |- ( ( ph /\ A R x ) -> A R A )
12	8, 11	exlimddv 1863	1 |- ( ph -> A R A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   class class class wbr 4653   dom cdm 5114   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-eu 2474  df-mo 2475  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-cnv 5122  df-co 5123  df-dm 5124  df-er 7742

This theorem is referenced by:  iserd  7768  erth  7791  iiner  7819  erinxp  7821  nqerid  9755  enqeq  9756  qusgrp  17649  sylow2alem1  18032  sylow2alem2  18033  sylow2a  18034  efginvrel2  18140  efgsrel  18147  efgcpbllemb  18168  frgp0  18173  frgpnabllem1  18276  frgpnabllem2  18277  pcophtb  22829  pi1xfrf  22853  pi1xfr  22855  pi1xfrcnvlem  22856  prtlem10  34150