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Theorem eqvrelthi 38311
Description: Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)
Hypotheses
Ref Expression
eqvrelthi.1 (𝜑 → EqvRel 𝑅)
eqvrelthi.2 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
eqvrelthi (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Proof of Theorem eqvrelthi
StepHypRef Expression
1 eqvrelthi.2 . 2 (𝜑𝐴𝑅𝐵)
2 eqvrelthi.1 . . 3 (𝜑 → EqvRel 𝑅)
32, 1eqvrelcl 38310 . . 3 (𝜑𝐴 ∈ dom 𝑅)
42, 3eqvrelth 38309 . 2 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
51, 4mpbid 231 1 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534   class class class wbr 5153  [cec 8732   EqvRel weqvrel 37893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ec 8736  df-refrel 38210  df-symrel 38242  df-trrel 38272  df-eqvrel 38283
This theorem is referenced by:  eqvreldisj  38312  eqvrelqsel  38314
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