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Theorem brdmqss 35953
Description: The domain quotient binary relation. (Contributed by Peter Mazsa, 17-Apr-2019.)
Assertion
Ref Expression
brdmqss ((𝐴𝑉𝑅𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴))

Proof of Theorem brdmqss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmqseq 35947 . . . 4 (𝑥 = 𝑅 → (dom 𝑥 / 𝑥) = (dom 𝑅 / 𝑅))
2 id 22 . . . 4 (𝑦 = 𝐴𝑦 = 𝐴)
31, 2eqeqan12d 2841 . . 3 ((𝑥 = 𝑅𝑦 = 𝐴) → ((dom 𝑥 / 𝑥) = 𝑦 ↔ (dom 𝑅 / 𝑅) = 𝐴))
4 df-dmqss 35945 . . 3 DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦}
53, 4brabga 5409 . 2 ((𝑅𝑊𝐴𝑉) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴))
65ancoms 462 1 ((𝐴𝑉𝑅𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115   class class class wbr 5053  dom cdm 5543   / cqs 8280   DomainQss cdmqss 35548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-cnv 5551  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-ec 8283  df-qs 8287  df-dmqss 35945
This theorem is referenced by:  brdmqssqs  35954  cnvepresdmqss  35958
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