Theorem cnvepresdmqss 38756

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	⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) DomainQss 𝐴 ↔ ¬ ∅ ∈ 𝐴))

Proof of Theorem cnvepresdmqss

Step	Hyp	Ref	Expression
1		cnvepresex 38374	. . 3 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V)
2		brdmqss 38749	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (◡ E ↾ 𝐴) ∈ V) → ((◡ E ↾ 𝐴) DomainQss 𝐴 ↔ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴))
3	1, 2	mpdan 687	. 2 ⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) DomainQss 𝐴 ↔ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴))
4		n0el3 38755	. 2 ⊢ (¬ ∅ ∈ 𝐴 ↔ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)
5	3, 4	bitr4di 289	1 ⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) DomainQss 𝐴 ↔ ¬ ∅ ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ∅c0 4282   class class class wbr 5093   E cep 5518  ^◡ccnv 5618  dom cdm 5619   ↾ cres 5621   / cqs 8627   DomainQss cdmqss 38251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-eprel 5519  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ec 8630  df-qs 8634  df-dmqss 38740

This theorem is referenced by: (None)