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Theorem cnvepresdmqss 38012
Description: The domain quotient binary relation of 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
cnvepresdmqss (𝐴𝑉 → (( E ↾ 𝐴) DomainQss 𝐴 ↔ ¬ ∅ ∈ 𝐴))

Proof of Theorem cnvepresdmqss
StepHypRef Expression
1 cnvepresex 37693 . . 3 (𝐴𝑉 → ( E ↾ 𝐴) ∈ V)
2 brdmqss 38006 . . 3 ((𝐴𝑉 ∧ ( E ↾ 𝐴) ∈ V) → (( E ↾ 𝐴) DomainQss 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴))
31, 2mpdan 684 . 2 (𝐴𝑉 → (( E ↾ 𝐴) DomainQss 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴))
4 n0el3 38011 . 2 (¬ ∅ ∈ 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)
53, 4bitr4di 289 1 (𝐴𝑉 → (( E ↾ 𝐴) DomainQss 𝐴 ↔ ¬ ∅ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1533  wcel 2098  Vcvv 3466  c0 4314   class class class wbr 5138   E cep 5569  ccnv 5665  dom cdm 5666  cres 5668   / cqs 8698   DomainQss cdmqss 37556
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-eprel 5570  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ec 8701  df-qs 8705  df-dmqss 37998
This theorem is referenced by: (None)
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