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Theorem erref 8766
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 (𝜑𝑅 Er 𝑋)
erref.2 (𝜑𝐴𝑋)
Assertion
Ref Expression
erref (𝜑𝐴𝑅𝐴)

Proof of Theorem erref
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 erref.2 . . . 4 (𝜑𝐴𝑋)
2 ersymb.1 . . . . 5 (𝜑𝑅 Er 𝑋)
3 erdm 8756 . . . . 5 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
42, 3syl 17 . . . 4 (𝜑 → dom 𝑅 = 𝑋)
51, 4eleqtrrd 2843 . . 3 (𝜑𝐴 ∈ dom 𝑅)
6 eldmg 5908 . . . 4 (𝐴𝑋 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
71, 6syl 17 . . 3 (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
85, 7mpbid 232 . 2 (𝜑 → ∃𝑥 𝐴𝑅𝑥)
92adantr 480 . . 3 ((𝜑𝐴𝑅𝑥) → 𝑅 Er 𝑋)
10 simpr 484 . . 3 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝑥)
119, 10, 10ertr4d 8765 . 2 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝐴)
128, 11exlimddv 1934 1 (𝜑𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wex 1778  wcel 2107   class class class wbr 5142  dom cdm 5684   Er wer 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-er 8746
This theorem is referenced by:  iserd  8772  ecref  8791  erth  8797  iiner  8830  erinxp  8832  nqerid  10974  enqeq  10975  qusgrp  19205  sylow2alem1  19636  sylow2alem2  19637  sylow2a  19638  efginvrel2  19746  efgsrel  19753  efgcpbllemb  19774  frgp0  19779  frgpnabllem1  19892  frgpnabllem2  19893  pcophtb  25063  pi1xfrf  25087  pi1xfr  25089  pi1xfrcnvlem  25090  prtlem10  38867  prjspner01  42640  prjspner1  42641
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