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Theorem eqvrelref 36366
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 36353 . . . 4 ( EqvRel 𝑅 → Rel 𝑅)
4 releldmb 5789 . . . 4 (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
52, 3, 43syl 18 . . 3 (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
61, 5mpbid 235 . 2 (𝜑 → ∃𝑥 𝐴𝑅𝑥)
72adantr 484 . . 3 ((𝜑𝐴𝑅𝑥) → EqvRel 𝑅)
8 simpr 488 . . 3 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝑥)
97, 8, 8eqvreltr4d 36365 . 2 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝐴)
106, 9exlimddv 1942 1 (𝜑𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1786  wcel 2114   class class class wbr 5030  dom cdm 5525  Rel wrel 5530   EqvRel weqvrel 35993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-refrel 36273  df-symrel 36301  df-trrel 36331  df-eqvrel 36341
This theorem is referenced by:  eqvrelth  36367
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