Theorem brdmqss 36738

Description: The domain quotient binary relation. (Contributed by Peter Mazsa, 17-Apr-2019.)

Assertion

Ref	Expression
brdmqss	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴))

Proof of Theorem brdmqss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmqseq 36732	. . . 4 ⊢ (𝑥 = 𝑅 → (dom 𝑥 / 𝑥) = (dom 𝑅 / 𝑅))
2		id 22	. . . 4 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
3	1, 2	eqeqan12d 2753	. . 3 ⊢ ((𝑥 = 𝑅 ∧ 𝑦 = 𝐴) → ((dom 𝑥 / 𝑥) = 𝑦 ↔ (dom 𝑅 / 𝑅) = 𝐴))
4		df-dmqss 36730	. . 3 ⊢ DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦}
5	3, 4	brabga 5448	. 2 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴))
6	5	ancoms 458	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1541   ∈ wcel 2109   class class class wbr 5078  dom cdm 5588   / cqs 8471   DomainQss cdmqss 36335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-cnv 5596  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-ec 8474  df-qs 8478  df-dmqss 36730

This theorem is referenced by:  brdmqssqs  36739  cnvepresdmqss  36743