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Mirrors > Home > MPE Home > Th. List > Mathboxes > clsneifv4 | Structured version Visualization version GIF version |
Description: Value of the closure (interior) function in terms of the neighborhoods (convergents) function. (Contributed by RP, 27-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) |
clsneifv.s | ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) |
Ref | Expression |
---|---|
clsneifv4 | ⊢ (𝜑 → (𝐾‘𝑆) = {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfin5 3943 | . 2 ⊢ (𝐵 ∩ (𝐾‘𝑆)) = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ (𝐾‘𝑆)} | |
2 | clsnei.o | . . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
3 | clsnei.p | . . . . . . 7 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) | |
4 | clsnei.d | . . . . . . 7 ⊢ 𝐷 = (𝑃‘𝐵) | |
5 | clsnei.f | . . . . . . 7 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
6 | clsnei.h | . . . . . . 7 ⊢ 𝐻 = (𝐹 ∘ 𝐷) | |
7 | clsnei.r | . . . . . . 7 ⊢ (𝜑 → 𝐾𝐻𝑁) | |
8 | 2, 3, 4, 5, 6, 7 | clsneikex 40454 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
9 | elmapi 8427 | . . . . . 6 ⊢ (𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐾:𝒫 𝐵⟶𝒫 𝐵) |
11 | clsneifv.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
12 | 10, 11 | ffvelrnd 6851 | . . . 4 ⊢ (𝜑 → (𝐾‘𝑆) ∈ 𝒫 𝐵) |
13 | 12 | elpwid 4549 | . . 3 ⊢ (𝜑 → (𝐾‘𝑆) ⊆ 𝐵) |
14 | sseqin2 4191 | . . 3 ⊢ ((𝐾‘𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐾‘𝑆)) = (𝐾‘𝑆)) | |
15 | 13, 14 | sylib 220 | . 2 ⊢ (𝜑 → (𝐵 ∩ (𝐾‘𝑆)) = (𝐾‘𝑆)) |
16 | 7 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐾𝐻𝑁) |
17 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
18 | 11 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑆 ∈ 𝒫 𝐵) |
19 | 2, 3, 4, 5, 6, 16, 17, 18 | clsneiel1 40456 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐾‘𝑆) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥))) |
20 | 19 | rabbidva 3478 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ (𝐾‘𝑆)} = {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)}) |
21 | 1, 15, 20 | 3eqtr3a 2880 | 1 ⊢ (𝜑 → (𝐾‘𝑆) = {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 class class class wbr 5065 ↦ cmpt 5145 ∘ ccom 5558 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 ↑m cmap 8405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-map 8407 |
This theorem is referenced by: (None) |
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