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Theorem dmqseq 35911
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 5769 . 2 (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆)
2 qseq12 8344 . 2 ((dom 𝑅 = dom 𝑆𝑅 = 𝑆) → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
31, 2mpancom 686 1 (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  dom cdm 5552   / cqs 8285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rex 3143  df-rab 3146  df-v 3495  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4465  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-cnv 5560  df-dm 5562  df-rn 5563  df-res 5564  df-ima 5565  df-ec 8288  df-qs 8292
This theorem is referenced by:  dmqseqi  35912  dmqseqd  35913  dmqseqeq1  35914  brdmqss  35917
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