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Theorem erref 7747
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 7737 . . . . 5 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
42, 3syl 17 . . . 4 (𝜑 → dom 𝑅 = 𝑋)
51, 4eleqtrrd 2702 . . 3 (𝜑𝐴 ∈ dom 𝑅)
6 eldmg 5308 . . . 4 (𝐴𝑋 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
71, 6syl 17 . . 3 (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
85, 7mpbid 222 . 2 (𝜑 → ∃𝑥 𝐴𝑅𝑥)
92adantr 481 . . 3 ((𝜑𝐴𝑅𝑥) → 𝑅 Er 𝑋)
10 simpr 477 . . 3 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝑥)
119, 10, 10ertr4d 7746 . 2 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝐴)
128, 11exlimddv 1861 1 (𝜑𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988   class class class wbr 4644  dom cdm 5104   Er wer 7724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-er 7727
This theorem is referenced by:  iserd  7753  erth  7776  iiner  7804  erinxp  7806  nqerid  9740  enqeq  9741  qusgrp  17630  sylow2alem1  18013  sylow2alem2  18014  sylow2a  18015  efginvrel2  18121  efgsrel  18128  efgcpbllemb  18149  frgp0  18154  frgpnabllem1  18257  frgpnabllem2  18258  pcophtb  22810  pi1xfrf  22834  pi1xfr  22836  pi1xfrcnvlem  22837  prtlem10  33969
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