![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
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 43430 | . . 3 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | clsneiel1 43435 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)) ↔ ¬ (𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑁‘𝑋))) |
10 | clsneiel.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
11 | 10 | elpwid 4606 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
12 | dfss4 4253 | . . . . 5 ⊢ (𝑆 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑆)) = 𝑆) | |
13 | 11, 12 | sylib 217 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ 𝑆)) = 𝑆) |
14 | 13 | eleq1d 2812 | . . 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 205 = wceq 1533 ∈ wcel 2098 {crab 3426 Vcvv 3468 ∖ cdif 3940 ⊆ wss 3943 𝒫 cpw 4597 class class class wbr 5141 ↦ cmpt 5224 ∘ ccom 5673 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 ↑m cmap 8822 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-map 8824 |
This theorem is referenced by: clsneifv3 43437 |
Copyright terms: Public domain | W3C validator |