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Theorem eqvrelref 37993
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.) (Revised by Peter Mazsa, 2-Jun-2019.)
Hypotheses
Ref Expression
eqvrelref.1 (𝜑 → EqvRel 𝑅)
eqvrelref.2 (𝜑𝐴 ∈ dom 𝑅)
Assertion
Ref Expression
eqvrelref (𝜑𝐴𝑅𝐴)

Proof of Theorem eqvrelref
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqvrelref.2 . . 3 (𝜑𝐴 ∈ dom 𝑅)
2 eqvrelref.1 . . . 4 (𝜑 → EqvRel 𝑅)
3 eqvrelrel 37980 . . . 4 ( EqvRel 𝑅 → Rel 𝑅)
4 releldmb 5939 . . . 4 (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
52, 3, 43syl 18 . . 3 (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
61, 5mpbid 231 . 2 (𝜑 → ∃𝑥 𝐴𝑅𝑥)
72adantr 480 . . 3 ((𝜑𝐴𝑅𝑥) → EqvRel 𝑅)
8 simpr 484 . . 3 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝑥)
97, 8, 8eqvreltr4d 37992 . 2 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝐴)
106, 9exlimddv 1930 1 (𝜑𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wex 1773  wcel 2098   class class class wbr 5141  dom cdm 5669  Rel wrel 5674   EqvRel weqvrel 37573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-refrel 37895  df-symrel 37927  df-trrel 37957  df-eqvrel 37968
This theorem is referenced by:  eqvrelth  37994
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