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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 3582 | . 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 38416 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑𝑚 𝐵)) |
10 | elmapi 7879 | . . . . . 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 6360 | . . . 4 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵) |
14 | 13 | elpwid 4170 | . . 3 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵) |
15 | sseqin2 3817 | . . 3 ⊢ ((𝑁‘𝑋) ⊆ 𝒫 𝐵 ↔ (𝒫 𝐵 ∩ (𝑁‘𝑋)) = (𝑁‘𝑋)) | |
16 | 14, 15 | sylib 208 | . 2 ⊢ (𝜑 → (𝒫 𝐵 ∩ (𝑁‘𝑋)) = (𝑁‘𝑋)) |
17 | 8 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑁𝐻𝑀) |
18 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵) |
19 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵) | |
20 | 2, 3, 4, 5, 6, 7, 17, 18, 19 | neicvgel1 38417 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑠 ∈ (𝑁‘𝑋) ↔ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋))) |
21 | 20 | rabbidva 3188 | . 2 ⊢ (𝜑 → {𝑠 ∈ 𝒫 𝐵 ∣ 𝑠 ∈ (𝑁‘𝑋)} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)}) |
22 | 1, 16, 21 | 3eqtr3a 2680 | 1 ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 class class class wbr 4653 ↦ cmpt 4729 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ↑𝑚 cmap 7857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 |
This theorem is referenced by: (None) |
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