Theorem erref 8721

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	⊢ (𝜑 → 𝑅 Er 𝑋)
erref.2	⊢ (𝜑 → 𝐴 ∈ 𝑋)

Assertion

Ref	Expression
erref	⊢ (𝜑 → 𝐴𝑅𝐴)

Proof of Theorem erref

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		erref.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
2		ersymb.1	. . . . 5 ⊢ (𝜑 → 𝑅 Er 𝑋)
3		erdm 8711	. . . . 5 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
4	2, 3	syl 18	. . . 4 ⊢ (𝜑 → dom 𝑅 = 𝑋)
5	1, 4	eleqtrrd 2865	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
6		eldmg 5886	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
7	1, 6	syl 18	. . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
8	5, 7	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑥 𝐴𝑅𝑥)
9	2	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝑅 Er 𝑋)
10		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝑥)
11	9, 10, 10	ertr4d 8720	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴)
12	8, 11	exlimddv 1968	1 ⊢ (𝜑 → 𝐴𝑅𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   class class class wbr 5107  dom cdm 5659   Er wer 8697

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-er 8700

This theorem is used by:  iserd  8727  ecref  8746  erth  8755  iiner  8793  erinxp  8795  nqerid  10946  enqeq  10947  qusgrp  19320  sylow2alem1  19750  sylow2alem2  19751  sylow2a  19752  efginvrel2  19860  efgsrel  19867  efgcpbllemb  19888  frgp0  19893  frgpnabllem1  20006  frgpnabllem2  20007  pcophtb  25263  pi1xfrf  25287  pi1xfr  25289  pi1xfrcnvlem  25290  prtlem10  39746  prjspner01  43479  prjspner1  43480  chnerlem1  47718  chner  47721