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Theorem dmqseq 38021
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 5896 . 2 (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆)
2 qseq12 8760 . 2 ((dom 𝑅 = dom 𝑆𝑅 = 𝑆) → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
31, 2mpancom 685 1 (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  dom cdm 5669   / cqs 8701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8704  df-qs 8708
This theorem is referenced by:  dmqseqi  38022  dmqseqd  38023  dmqseqeq1  38024  brdmqss  38027
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