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Theorem brdmqss 38150
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 38144 . . . 4 (𝑥 = 𝑅 → (dom 𝑥 / 𝑥) = (dom 𝑅 / 𝑅))
2 id 22 . . . 4 (𝑦 = 𝐴𝑦 = 𝐴)
31, 2eqeqan12d 2742 . . 3 ((𝑥 = 𝑅𝑦 = 𝐴) → ((dom 𝑥 / 𝑥) = 𝑦 ↔ (dom 𝑅 / 𝑅) = 𝐴))
4 df-dmqss 38142 . . 3 DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦}
53, 4brabga 5540 . 2 ((𝑅𝑊𝐴𝑉) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴))
65ancoms 457 1 ((𝐴𝑉𝑅𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098   class class class wbr 5152  dom cdm 5682   / cqs 8730   DomainQss cdmqss 37704
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
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 2706  df-cleq 2720  df-clel 2806  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ec 8733  df-qs 8737  df-dmqss 38142
This theorem is referenced by:  brdmqssqs  38151  cnvepresdmqss  38156  brparts2  38276
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