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Theorem dmec2d 37777
Description: Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 8774). (Contributed by Peter Mazsa, 12-Oct-2018.)
Hypothesis
Ref Expression
dmec2d.1 (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)
Assertion
Ref Expression
dmec2d (𝜑 → (𝐵 ∈ dom 𝑅𝐶 ∈ dom 𝑅))

Proof of Theorem dmec2d
StepHypRef Expression
1 eqidd 2729 . 2 (𝜑 → dom 𝑅 = dom 𝑅)
2 dmec2d.1 . 2 (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)
31, 2dmecd 37776 1 (𝜑 → (𝐵 ∈ dom 𝑅𝐶 ∈ dom 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  dom cdm 5678  [cec 8723
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 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8727
This theorem is referenced by:  eqvrelth  38083
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