Theorem erref 6607

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 6597	. . . . 5 |- ( R Er X -> dom R = X )
4	2, 3	syl 14	. . . 4 |- ( ph -> dom R = X )
5	1, 4	eleqtrrd 2273	. . 3 |- ( ph -> A e. dom R )
6		eldmg 4857	. . . 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 6606	. 2 |- ( ( ph /\ A R x ) -> A R A )
12	8, 11	exlimddv 1910	1 |- ( ph -> A R A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2164   class class class wbr 4029   dom cdm 4659   Er wer 6584

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-er 6587

This theorem is referenced by:  iserd  6613  erth  6633  iinerm  6661  erinxp  6663  qusgrp  13302