Theorem erref 6521

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 6511	. . . . 5 |- ( R Er X -> dom R = X )
4	2, 3	syl 14	. . . 4 |- ( ph -> dom R = X )
5	1, 4	eleqtrrd 2246	. . 3 |- ( ph -> A e. dom R )
6		eldmg 4799	. . . 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 146	. 2 |- ( ph -> E. x A R x )
9	2	adantr 274	. . 3 |- ( ( ph /\ A R x ) -> R Er X )
10		simpr 109	. . 3 |- ( ( ph /\ A R x ) -> A R x )
11	9, 10, 10	ertr4d 6520	. 2 |- ( ( ph /\ A R x ) -> A R A )
12	8, 11	exlimddv 1886	1 |- ( ph -> A R A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   class class class wbr 3982   dom cdm 4604   Er wer 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-er 6501

This theorem is referenced by:  iserd  6527  erth  6545  iinerm  6573  erinxp  6575