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Theorem eqvrelref 34900
 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 34887 . . . 4 ( EqvRel 𝑅 → Rel 𝑅)
4 releldmb 5593 . . . 4 (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
52, 3, 43syl 18 . . 3 (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
61, 5mpbid 224 . 2 (𝜑 → ∃𝑥 𝐴𝑅𝑥)
72adantr 474 . . 3 ((𝜑𝐴𝑅𝑥) → EqvRel 𝑅)
8 simpr 479 . . 3 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝑥)
97, 8, 8eqvreltr4d 34899 . 2 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝐴)
106, 9exlimddv 2036 1 (𝜑𝐴𝑅𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∃wex 1880   ∈ wcel 2166   class class class wbr 4873  dom cdm 5342  Rel wrel 5347   EqvRel weqvrel 34541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-refrel 34810  df-symrel 34838  df-trrel 34868  df-eqvrel 34878 This theorem is referenced by:  eqvrelth  34901
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