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Theorem cnvepresdmqss 37143
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 36824 . . 3 (𝐴𝑉 → ( E ↾ 𝐴) ∈ V)
2 brdmqss 37137 . . 3 ((𝐴𝑉 ∧ ( E ↾ 𝐴) ∈ V) → (( E ↾ 𝐴) DomainQss 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴))
31, 2mpdan 686 . 2 (𝐴𝑉 → (( E ↾ 𝐴) DomainQss 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴))
4 n0el3 37142 . 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 1542  wcel 2107  Vcvv 3448  c0 4287   class class class wbr 5110   E cep 5541  ccnv 5637  dom cdm 5638  cres 5640   / cqs 8654   DomainQss cdmqss 36686
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-eprel 5542  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ec 8657  df-qs 8661  df-dmqss 37129
This theorem is referenced by: (None)
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