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Theorem cnvepresdmqs 36411
Description: The domain quotient predicate for the restricted converse epsilon relation is equivalent to the negated elementhood of the empty set in the restriction. (Contributed by Peter Mazsa, 14-Aug-2021.)
Assertion
Ref Expression
cnvepresdmqs (( E ↾ 𝐴) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴)

Proof of Theorem cnvepresdmqs
StepHypRef Expression
1 df-dmqs 36398 . 2 (( E ↾ 𝐴) DomainQs 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)
2 n0el3 36409 . 2 (¬ ∅ ∈ 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)
31, 2bitr4i 281 1 (( E ↾ 𝐴) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209   = wceq 1542  wcel 2114  c0 4212   E cep 5434  ccnv 5525  dom cdm 5526  cres 5528   / cqs 8322   DomainQs wdmqs 36003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-br 5032  df-opab 5094  df-eprel 5435  df-xp 5532  df-rel 5533  df-cnv 5534  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-ec 8325  df-qs 8329  df-dmqs 36398
This theorem is referenced by: (None)
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