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Theorem dmqseq 38112
Description: Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.)
Assertion
Ref Expression
dmqseq (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))

Proof of Theorem dmqseq
StepHypRef Expression
1 dmeq 5906 . 2 (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆)
2 qseq12 8785 . 2 ((dom 𝑅 = dom 𝑆𝑅 = 𝑆) → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
31, 2mpancom 687 1 (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  dom cdm 5678   / cqs 8724
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-ext 2699
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-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  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-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8727  df-qs 8731
This theorem is referenced by:  dmqseqi  38113  dmqseqd  38114  dmqseqeq1  38115  brdmqss  38118
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