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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvepresdmqss | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
cnvepresdmqss | ⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) DomainQss 𝐴 ↔ ¬ ∅ ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvepresex 37858 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V) | |
2 | brdmqss 38170 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (◡ E ↾ 𝐴) ∈ V) → ((◡ E ↾ 𝐴) DomainQss 𝐴 ↔ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)) | |
3 | 1, 2 | mpdan 685 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) DomainQss 𝐴 ↔ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)) |
4 | n0el3 38175 | . 2 ⊢ (¬ ∅ ∈ 𝐴 ↔ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴) | |
5 | 3, 4 | bitr4di 288 | 1 ⊢ (𝐴 ∈ 𝑉 → ((◡ 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) |
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