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Mirrors > Home > MPE Home > Th. List > Mathboxes > neicvgel2 | Structured version Visualization version GIF version |
Description: The complement of a subset being an element of a neighborhood at a point is equivalent to that subset not being a element of the convergent at that point. (Contributed by RP, 12-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) |
neicvgel.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
neicvgel.s | ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) |
Ref | Expression |
---|---|
neicvgel2 | ⊢ (𝜑 → ((𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋) ↔ ¬ 𝑆 ∈ (𝑀‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neicvg.o | . . 3 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
2 | neicvg.p | . . 3 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) | |
3 | neicvg.d | . . 3 ⊢ 𝐷 = (𝑃‘𝐵) | |
4 | neicvg.f | . . 3 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
5 | neicvg.g | . . 3 ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) | |
6 | neicvg.h | . . 3 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) | |
7 | neicvg.r | . . 3 ⊢ (𝜑 → 𝑁𝐻𝑀) | |
8 | neicvgel.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | 3, 6, 7 | neicvgrcomplex 40341 | . . 3 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | neicvgel1 40347 | . 2 ⊢ (𝜑 → ((𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋) ↔ ¬ (𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑀‘𝑋))) |
11 | neicvgel.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
12 | 11 | elpwid 4549 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
13 | dfss4 4232 | . . . . 5 ⊢ (𝑆 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑆)) = 𝑆) | |
14 | 12, 13 | sylib 219 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ 𝑆)) = 𝑆) |
15 | 14 | eleq1d 2894 | . . 3 ⊢ (𝜑 → ((𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑀‘𝑋) ↔ 𝑆 ∈ (𝑀‘𝑋))) |
16 | 15 | notbid 319 | . 2 ⊢ (𝜑 → (¬ (𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑀‘𝑋) ↔ ¬ 𝑆 ∈ (𝑀‘𝑋))) |
17 | 10, 16 | bitrd 280 | 1 ⊢ (𝜑 → ((𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋) ↔ ¬ 𝑆 ∈ (𝑀‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 𝒫 cpw 4535 class class class wbr 5057 ↦ cmpt 5137 ∘ ccom 5552 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ↑m cmap 8395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-map 8397 |
This theorem is referenced by: (None) |
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