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Theorem erref 7622
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 7612 . . . . 5 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
42, 3syl 17 . . . 4 (𝜑 → dom 𝑅 = 𝑋)
51, 4eleqtrrd 2686 . . 3 (𝜑𝐴 ∈ dom 𝑅)
6 eldmg 5224 . . . 4 (𝐴𝑋 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
71, 6syl 17 . . 3 (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
85, 7mpbid 220 . 2 (𝜑 → ∃𝑥 𝐴𝑅𝑥)
92adantr 479 . . 3 ((𝜑𝐴𝑅𝑥) → 𝑅 Er 𝑋)
10 simpr 475 . . 3 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝑥)
119, 10, 10ertr4d 7621 . 2 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝐴)
128, 11exlimddv 1848 1 (𝜑𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1975   class class class wbr 4573  dom cdm 5024   Er wer 7599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-opab 4634  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-er 7602
This theorem is referenced by:  iserd  7628  erth  7651  iiner  7679  erinxp  7681  nqerid  9607  enqeq  9608  qusgrp  17414  sylow2alem1  17797  sylow2alem2  17798  sylow2a  17799  efginvrel2  17905  efgsrel  17912  efgcpbllemb  17933  frgp0  17938  frgpnabllem1  18041  frgpnabllem2  18042  pcophtb  22564  pi1xfrf  22588  pi1xfr  22590  pi1xfrcnvlem  22591  prtlem10  32967
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