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Theorem brdmqssqs 38175
Description: If 𝐴 and 𝑅 are sets, the domain quotient binary relation and the domain quotient predicate are the same. (Contributed by Peter Mazsa, 14-Aug-2021.)
Assertion
Ref Expression
brdmqssqs ((𝐴𝑉𝑅𝑊) → (𝑅 DomainQss 𝐴𝑅 DomainQs 𝐴))

Proof of Theorem brdmqssqs
StepHypRef Expression
1 brdmqss 38174 . 2 ((𝐴𝑉𝑅𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴))
2 df-dmqs 38167 . 2 (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
31, 2bitr4di 288 1 ((𝐴𝑉𝑅𝑊) → (𝑅 DomainQss 𝐴𝑅 DomainQs 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098   class class class wbr 5143  dom cdm 5672   / cqs 8722   DomainQss cdmqss 37728   DomainQs wdmqs 37729
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 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
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 2703  df-cleq 2717  df-clel 2802  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8725  df-qs 8729  df-dmqss 38166  df-dmqs 38167
This theorem is referenced by:  brerser  38205  brpartspart  38301
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