Theorem dmqseqd 39356

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 39354	. 2 ⊢ (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
3	1, 2	syl 18	1 ⊢ (𝜑 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))

Syntax hints:   → wi 4   = wceq 1570  dom cdm 5663   / cqs 8694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ec 8697  df-qs 8701

This theorem is referenced by:  releldmqs  39373  releldmqscoss  39375