Theorem erref 6800

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 6790	. . . . 5 |- ( R Er X -> dom R = X )
4	2, 3	syl 14	. . . 4 |- ( ph -> dom R = X )
5	1, 4	eleqtrrd 2314	. . 3 |- ( ph -> A e. dom R )
6		eldmg 4956	. . . 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 6799	. 2 |- ( ( ph /\ A R x ) -> A R A )
12	8, 11	exlimddv 1950	1 |- ( ph -> A R A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   class class class wbr 4114   dom cdm 4754   Er wer 6777

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-er 6780

This theorem is referenced by:  iserd  6806  erth  6826  iinerm  6854  erinxp  6856  qusgrp  13985