Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cnvepresdmqss Structured version   Visualization version   GIF version

Theorem cnvepresdmqss 38176
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 37858 . . 3 (𝐴𝑉 → ( E ↾ 𝐴) ∈ V)
2 brdmqss 38170 . . 3 ((𝐴𝑉 ∧ ( E ↾ 𝐴) ∈ V) → (( E ↾ 𝐴) DomainQss 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴))
31, 2mpdan 685 . 2 (𝐴𝑉 → (( E ↾ 𝐴) DomainQss 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴))
4 n0el3 38175 . 2 (¬ ∅ ∈ 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)
53, 4bitr4di 288 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 3463  c0 4319   class class class wbr 5144   E cep 5576  ccnv 5672  dom cdm 5673  cres 5675   / cqs 8717   DomainQss cdmqss 37724
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 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-eprel 5577  df-xp 5679  df-rel 5680  df-cnv 5681  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ec 8720  df-qs 8724  df-dmqss 38162
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator