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Theorem erref 7649
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 7639 . . . . 5 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
42, 3syl 17 . . . 4 (𝜑 → dom 𝑅 = 𝑋)
51, 4eleqtrrd 2691 . . 3 (𝜑𝐴 ∈ dom 𝑅)
6 eldmg 5241 . . . 4 (𝐴𝑋 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
71, 6syl 17 . . 3 (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
85, 7mpbid 221 . 2 (𝜑 → ∃𝑥 𝐴𝑅𝑥)
92adantr 480 . . 3 ((𝜑𝐴𝑅𝑥) → 𝑅 Er 𝑋)
10 simpr 476 . . 3 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝑥)
119, 10, 10ertr4d 7648 . 2 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝐴)
128, 11exlimddv 1850 1 (𝜑𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wex 1695  wcel 1977   class class class wbr 4583  dom cdm 5038   Er wer 7626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-er 7629
This theorem is referenced by:  iserd  7655  erth  7678  iiner  7706  erinxp  7708  nqerid  9634  enqeq  9635  qusgrp  17472  sylow2alem1  17855  sylow2alem2  17856  sylow2a  17857  efginvrel2  17963  efgsrel  17970  efgcpbllemb  17991  frgp0  17996  frgpnabllem1  18099  frgpnabllem2  18100  pcophtb  22637  pi1xfrf  22661  pi1xfr  22663  pi1xfrcnvlem  22664  prtlem10  33168
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