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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 41408 | . . 3 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | neicvgel1 41414 | . 2 ⊢ (𝜑 → ((𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋) ↔ ¬ (𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑀‘𝑋))) |
11 | neicvgel.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
12 | 11 | elpwid 4529 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
13 | dfss4 4178 | . . . . 5 ⊢ (𝑆 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑆)) = 𝑆) | |
14 | 12, 13 | sylib 221 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ 𝑆)) = 𝑆) |
15 | 14 | eleq1d 2822 | . . 3 ⊢ (𝜑 → ((𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑀‘𝑋) ↔ 𝑆 ∈ (𝑀‘𝑋))) |
16 | 15 | notbid 321 | . 2 ⊢ (𝜑 → (¬ (𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑀‘𝑋) ↔ ¬ 𝑆 ∈ (𝑀‘𝑋))) |
17 | 10, 16 | bitrd 282 | 1 ⊢ (𝜑 → ((𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋) ↔ ¬ 𝑆 ∈ (𝑀‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 {crab 3065 Vcvv 3413 ∖ cdif 3868 ⊆ wss 3871 𝒫 cpw 4518 class class class wbr 5058 ↦ cmpt 5140 ∘ ccom 5560 ‘cfv 6385 (class class class)co 7218 ∈ cmpo 7220 ↑m cmap 8513 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5184 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-op 4553 df-uni 4825 df-iun 4911 df-br 5059 df-opab 5121 df-mpt 5141 df-id 5460 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-ov 7221 df-oprab 7222 df-mpo 7223 df-1st 7766 df-2nd 7767 df-map 8515 |
This theorem is referenced by: (None) |
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