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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvepresdmqs | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
cnvepresdmqs | ⊢ ((◡ E ↾ 𝐴) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴) |
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) |
Colors of variables: wff setvar class |
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) |
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