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Mirrors > Home > MPE Home > Th. List > Mathboxes > clsneiel2 | Structured version Visualization version GIF version |
Description: If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then membership in the closure of the complement of a subset is equivalent to the subset not being a neighborhood of the point. (Contributed by RP, 7-Jun-2021.) |
Ref | Expression |
---|---|
clsnei.o | ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) |
clsnei.p | ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) |
clsnei.d | ⊢ 𝐷 = (𝑃‘𝐵) |
clsnei.f | ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) |
clsnei.h | ⊢ 𝐻 = (𝐹 ∘ 𝐷) |
clsnei.r | ⊢ (𝜑 → 𝐾𝐻𝑁) |
clsneiel.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
clsneiel.s | ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) |
Ref | Expression |
---|---|
clsneiel2 | ⊢ (𝜑 → (𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)) ↔ ¬ 𝑆 ∈ (𝑁‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clsnei.o | . . 3 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
2 | clsnei.p | . . 3 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) | |
3 | clsnei.d | . . 3 ⊢ 𝐷 = (𝑃‘𝐵) | |
4 | clsnei.f | . . 3 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
5 | clsnei.h | . . 3 ⊢ 𝐻 = (𝐹 ∘ 𝐷) | |
6 | clsnei.r | . . 3 ⊢ (𝜑 → 𝐾𝐻𝑁) | |
7 | clsneiel.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | 3, 5, 6 | clsneircomplex 44092 | . . 3 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | clsneiel1 44097 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)) ↔ ¬ (𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑁‘𝑋))) |
10 | clsneiel.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
11 | 10 | elpwid 4613 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
12 | dfss4 4274 | . . . . 5 ⊢ (𝑆 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑆)) = 𝑆) | |
13 | 11, 12 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ 𝑆)) = 𝑆) |
14 | 13 | eleq1d 2823 | . . 3 ⊢ (𝜑 → ((𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑁‘𝑋) ↔ 𝑆 ∈ (𝑁‘𝑋))) |
15 | 14 | notbid 318 | . 2 ⊢ (𝜑 → (¬ (𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑁‘𝑋) ↔ ¬ 𝑆 ∈ (𝑁‘𝑋))) |
16 | 9, 15 | bitrd 279 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)) ↔ ¬ 𝑆 ∈ (𝑁‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 {crab 3432 Vcvv 3477 ∖ cdif 3959 ⊆ wss 3962 𝒫 cpw 4604 class class class wbr 5147 ↦ cmpt 5230 ∘ ccom 5692 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 ↑m cmap 8864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-map 8866 |
This theorem is referenced by: clsneifv3 44099 |
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