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Theorem cnvepresdmqss 36387
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 36092 . . 3 (𝐴𝑉 → ( E ↾ 𝐴) ∈ V)
2 brdmqss 36382 . . 3 ((𝐴𝑉 ∧ ( E ↾ 𝐴) ∈ V) → (( E ↾ 𝐴) DomainQss 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴))
31, 2mpdan 687 . 2 (𝐴𝑉 → (( E ↾ 𝐴) DomainQss 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴))
4 n0el3 36386 . 2 (¬ ∅ ∈ 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)
53, 4bitr4di 292 1 (𝐴𝑉 → (( E ↾ 𝐴) DomainQss 𝐴 ↔ ¬ ∅ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1542  wcel 2114  Vcvv 3398  c0 4211   class class class wbr 5030   E cep 5433  ccnv 5524  dom cdm 5525  cres 5527   / cqs 8319   DomainQss cdmqss 35979
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 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
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 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-eprel 5434  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ec 8322  df-qs 8326  df-dmqss 36374
This theorem is referenced by: (None)
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