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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > neicvgfv | Structured version Visualization version GIF version |
Description: The value of the neighborhoods (convergents) in terms of the the convergents (neighborhoods) function. (Contributed by RP, 27-Jun-2021.) |
Ref | Expression |
---|---|
neicvg.o | ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) |
neicvg.p | ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) |
neicvg.d | ⊢ 𝐷 = (𝑃‘𝐵) |
neicvg.f | ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) |
neicvg.g | ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) |
neicvg.h | ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) |
neicvg.r | ⊢ (𝜑 → 𝑁𝐻𝑀) |
neicvgfv.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
neicvgfv | ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfin5 3806 | . 2 ⊢ (𝒫 𝐵 ∩ (𝑁‘𝑋)) = {𝑠 ∈ 𝒫 𝐵 ∣ 𝑠 ∈ (𝑁‘𝑋)} | |
2 | neicvg.o | . . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
3 | neicvg.p | . . . . . . 7 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) | |
4 | neicvg.d | . . . . . . 7 ⊢ 𝐷 = (𝑃‘𝐵) | |
5 | neicvg.f | . . . . . . 7 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
6 | neicvg.g | . . . . . . 7 ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) | |
7 | neicvg.h | . . . . . . 7 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) | |
8 | neicvg.r | . . . . . . 7 ⊢ (𝜑 → 𝑁𝐻𝑀) | |
9 | 2, 3, 4, 5, 6, 7, 8 | neicvgnex 39249 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑𝑚 𝐵)) |
10 | elmapi 8144 | . . . . . 6 ⊢ (𝑁 ∈ (𝒫 𝒫 𝐵 ↑𝑚 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁:𝐵⟶𝒫 𝒫 𝐵) |
12 | neicvgfv.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
13 | 11, 12 | ffvelrnd 6609 | . . . 4 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵) |
14 | 13 | elpwid 4390 | . . 3 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵) |
15 | sseqin2 4044 | . . 3 ⊢ ((𝑁‘𝑋) ⊆ 𝒫 𝐵 ↔ (𝒫 𝐵 ∩ (𝑁‘𝑋)) = (𝑁‘𝑋)) | |
16 | 14, 15 | sylib 210 | . 2 ⊢ (𝜑 → (𝒫 𝐵 ∩ (𝑁‘𝑋)) = (𝑁‘𝑋)) |
17 | 8 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑁𝐻𝑀) |
18 | 12 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵) |
19 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵) | |
20 | 2, 3, 4, 5, 6, 7, 17, 18, 19 | neicvgel1 39250 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑠 ∈ (𝑁‘𝑋) ↔ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋))) |
21 | 20 | rabbidva 3401 | . 2 ⊢ (𝜑 → {𝑠 ∈ 𝒫 𝐵 ∣ 𝑠 ∈ (𝑁‘𝑋)} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)}) |
22 | 1, 16, 21 | 3eqtr3a 2885 | 1 ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 {crab 3121 Vcvv 3414 ∖ cdif 3795 ∩ cin 3797 ⊆ wss 3798 𝒫 cpw 4378 class class class wbr 4873 ↦ cmpt 4952 ∘ ccom 5346 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 ↦ cmpt2 6907 ↑𝑚 cmap 8122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-map 8124 |
This theorem is referenced by: (None) |
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