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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 3777 | . 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 39186 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
9 | elmapi 8117 | . . . . . 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 6586 | . . . 4 ⊢ (𝜑 → (𝐾‘𝑆) ∈ 𝒫 𝐵) |
13 | 12 | elpwid 4361 | . . 3 ⊢ (𝜑 → (𝐾‘𝑆) ⊆ 𝐵) |
14 | sseqin2 4015 | . . 3 ⊢ ((𝐾‘𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐾‘𝑆)) = (𝐾‘𝑆)) | |
15 | 13, 14 | sylib 210 | . 2 ⊢ (𝜑 → (𝐵 ∩ (𝐾‘𝑆)) = (𝐾‘𝑆)) |
16 | 7 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐾𝐻𝑁) |
17 | simpr 478 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
18 | 11 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑆 ∈ 𝒫 𝐵) |
19 | 2, 3, 4, 5, 6, 16, 17, 18 | clsneiel1 39188 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐾‘𝑆) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥))) |
20 | 19 | rabbidva 3372 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ (𝐾‘𝑆)} = {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)}) |
21 | 1, 15, 20 | 3eqtr3a 2857 | 1 ⊢ (𝜑 → (𝐾‘𝑆) = {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 {crab 3093 Vcvv 3385 ∖ cdif 3766 ∩ cin 3768 ⊆ wss 3769 𝒫 cpw 4349 class class class wbr 4843 ↦ cmpt 4922 ∘ ccom 5316 ⟶wf 6097 ‘cfv 6101 (class class class)co 6878 ↦ cmpt2 6880 ↑𝑚 cmap 8095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1st 7401 df-2nd 7402 df-map 8097 |
This theorem is referenced by: (None) |
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