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

Step	Hyp	Ref	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)
3	1, 2	bitr4i 281	1 ⊢ ((◡ E ↾ 𝐴) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴)

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)