Theorem dmqseq 38641

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 5914	. 2 ⊢ (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆)
2		qseq12 8806	. 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 1540  dom cdm 5685   / cqs 8744

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-qs 8751

This theorem is referenced by:  dmqseqi  38642  dmqseqd  38643  dmqseqeq1  38644  brdmqss  38647