Theorem dmqseqd 38118

Description: Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.)

Hypothesis

Ref	Expression
dmqseqd.1	⊢ (𝜑 → 𝑅 = 𝑆)

Assertion

Ref	Expression
dmqseqd	⊢ (𝜑 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))

Proof of Theorem dmqseqd

Step	Hyp	Ref	Expression
1		dmqseqd.1	. 2 ⊢ (𝜑 → 𝑅 = 𝑆)
2		dmqseq 38116	. 2 ⊢ (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
3	1, 2	syl 17	1 ⊢ (𝜑 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))

Syntax hints:   → wi 4   = wceq 1533  dom cdm 5680   / cqs 8728

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 2698

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-cnv 5688  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-ec 8731  df-qs 8735

This theorem is referenced by:  releldmqs  38134  releldmqscoss  38136