Theorem erref 6827

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 6817	. . . . 5 |- ( R Er X -> dom R = X )
4	2, 3	syl 14	. . . 4 |- ( ph -> dom R = X )
5	1, 4	eleqtrrd 2318	. . 3 |- ( ph -> A e. dom R )
6		eldmg 4976	. . . 4 |- ( A e. X -> ( A e. dom R <-> E. x A R x ) )
7	1, 6	syl 14	. . 3 |- ( ph -> ( A e. dom R <-> E. x A R x ) )
8	5, 7	mpbid 147	. 2 |- ( ph -> E. x A R x )
9	2	adantr 276	. . 3 |- ( ( ph /\ A R x ) -> R Er X )
10		simpr 110	. . 3 |- ( ( ph /\ A R x ) -> A R x )
11	9, 10, 10	ertr4d 6826	. 2 |- ( ( ph /\ A R x ) -> A R A )
12	8, 11	exlimddv 1954	1 |- ( ph -> A R A )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   class class class wbr 4130   dom cdm 4774   Er wer 6804

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-er 6807

This theorem is used by:  iserd  6833  erth  6853  iinerm  6881  erinxp  6883  qusgrp  14035