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Theorem brdmqssqs 37454
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 37453 . 2 ((𝐴𝑉𝑅𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴))
2 df-dmqs 37446 . 2 (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
31, 2bitr4di 289 1 ((𝐴𝑉𝑅𝑊) → (𝑅 DomainQss 𝐴𝑅 DomainQs 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   class class class wbr 5146  dom cdm 5674   / cqs 8697   DomainQss cdmqss 37003   DomainQs wdmqs 37004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-br 5147  df-opab 5209  df-cnv 5682  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-ec 8700  df-qs 8704  df-dmqss 37445  df-dmqs 37446
This theorem is referenced by:  brerser  37484  brpartspart  37580
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