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Mirrors > Home > MPE Home > Th. List > Mathboxes > neicvgnvo | Structured version Visualization version GIF version |
Description: If neighborhood and convergent functions are related by operator 𝐻, it is its own converse function. (Contributed by RP, 11-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) |
Ref | Expression |
---|---|
neicvgnvo | ⊢ (𝜑 → ◡𝐻 = 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neicvg.h | . . . . 5 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) | |
2 | 1 | cnveqi 5709 | . . . 4 ⊢ ◡𝐻 = ◡(𝐹 ∘ (𝐷 ∘ 𝐺)) |
3 | cnvco 5720 | . . . 4 ⊢ ◡(𝐹 ∘ (𝐷 ∘ 𝐺)) = (◡(𝐷 ∘ 𝐺) ∘ ◡𝐹) | |
4 | cnvco 5720 | . . . . 5 ⊢ ◡(𝐷 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐷) | |
5 | 4 | coeq1i 5694 | . . . 4 ⊢ (◡(𝐷 ∘ 𝐺) ∘ ◡𝐹) = ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹) |
6 | 2, 3, 5 | 3eqtri 2825 | . . 3 ⊢ ◡𝐻 = ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹) |
7 | neicvg.o | . . . . . 6 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
8 | neicvg.d | . . . . . . 7 ⊢ 𝐷 = (𝑃‘𝐵) | |
9 | neicvg.r | . . . . . . 7 ⊢ (𝜑 → 𝑁𝐻𝑀) | |
10 | 8, 1, 9 | neicvgbex 40815 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | 10 | pwexd 5245 | . . . . . 6 ⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
12 | neicvg.g | . . . . . 6 ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) | |
13 | neicvg.f | . . . . . 6 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
14 | 7, 10, 11, 12, 13 | fsovcnvd 40715 | . . . . 5 ⊢ (𝜑 → ◡𝐺 = 𝐹) |
15 | neicvg.p | . . . . . 6 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) | |
16 | 15, 8, 10 | dssmapnvod 40721 | . . . . 5 ⊢ (𝜑 → ◡𝐷 = 𝐷) |
17 | 14, 16 | coeq12d 5699 | . . . 4 ⊢ (𝜑 → (◡𝐺 ∘ ◡𝐷) = (𝐹 ∘ 𝐷)) |
18 | 7, 11, 10, 13, 12 | fsovcnvd 40715 | . . . 4 ⊢ (𝜑 → ◡𝐹 = 𝐺) |
19 | 17, 18 | coeq12d 5699 | . . 3 ⊢ (𝜑 → ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹) = ((𝐹 ∘ 𝐷) ∘ 𝐺)) |
20 | 6, 19 | syl5eq 2845 | . 2 ⊢ (𝜑 → ◡𝐻 = ((𝐹 ∘ 𝐷) ∘ 𝐺)) |
21 | coass 6085 | . . 3 ⊢ ((𝐹 ∘ 𝐷) ∘ 𝐺) = (𝐹 ∘ (𝐷 ∘ 𝐺)) | |
22 | 21, 1 | eqtr4i 2824 | . 2 ⊢ ((𝐹 ∘ 𝐷) ∘ 𝐺) = 𝐻 |
23 | 20, 22 | eqtrdi 2849 | 1 ⊢ (𝜑 → ◡𝐻 = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ∖ cdif 3878 𝒫 cpw 4497 class class class wbr 5030 ↦ cmpt 5110 ◡ccnv 5518 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ↑m cmap 8389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 |
This theorem is referenced by: neicvgnvor 40819 |
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