Theorem dmqseq 38676

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

Step	Hyp	Ref	Expression
1		dmeq 5843	. 2 ⊢ (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆)
2		qseq12 8686	. 2 ⊢ ((dom 𝑅 = dom 𝑆 ∧ 𝑅 = 𝑆) → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
3	1, 2	mpancom 688	1 ⊢ (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))

Syntax hints:   → wi 4   = wceq 1541  dom cdm 5616   / cqs 8621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ec 8624  df-qs 8628

This theorem is referenced by:  dmqseqi  38677  dmqseqd  38678  dmqseqeq1  38679  brdmqss  38682