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Theorem dmqseq 38622
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 5917 . 2 (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆)
2 qseq12 8805 . 2 ((dom 𝑅 = dom 𝑆𝑅 = 𝑆) → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
31, 2mpancom 688 1 (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  dom cdm 5689   / cqs 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-qs 8750
This theorem is referenced by:  dmqseqi  38623  dmqseqd  38624  dmqseqeq1  38625  brdmqss  38628
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