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Mirrors > Home > MPE Home > Th. List > Mathboxes > clsneicnv | Structured version Visualization version GIF version |
Description: If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then the converse of the operator is known. (Contributed by RP, 5-Jun-2021.) |
Ref | Expression |
---|---|
clsnei.o | ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) |
clsnei.p | ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) |
clsnei.d | ⊢ 𝐷 = (𝑃‘𝐵) |
clsnei.f | ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) |
clsnei.h | ⊢ 𝐻 = (𝐹 ∘ 𝐷) |
clsnei.r | ⊢ (𝜑 → 𝐾𝐻𝑁) |
Ref | Expression |
---|---|
clsneicnv | ⊢ (𝜑 → ◡𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clsnei.h | . . . 4 ⊢ 𝐻 = (𝐹 ∘ 𝐷) | |
2 | 1 | cnveqi 5444 | . . 3 ⊢ ◡𝐻 = ◡(𝐹 ∘ 𝐷) |
3 | cnvco 5455 | . . 3 ⊢ ◡(𝐹 ∘ 𝐷) = (◡𝐷 ∘ ◡𝐹) | |
4 | 2, 3 | eqtri 2774 | . 2 ⊢ ◡𝐻 = (◡𝐷 ∘ ◡𝐹) |
5 | clsnei.d | . . . 4 ⊢ 𝐷 = (𝑃‘𝐵) | |
6 | clsnei.r | . . . 4 ⊢ (𝜑 → 𝐾𝐻𝑁) | |
7 | 5, 1, 6 | clsneibex 38894 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
8 | clsnei.p | . . . . 5 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) | |
9 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐵 ∈ V) | |
10 | 8, 5, 9 | dssmapnvod 38808 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → ◡𝐷 = 𝐷) |
11 | clsnei.o | . . . . 5 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
12 | pwexg 4991 | . . . . . 6 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V) | |
13 | 12 | adantl 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝒫 𝐵 ∈ V) |
14 | clsnei.f | . . . . 5 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
15 | eqid 2752 | . . . . 5 ⊢ (𝐵𝑂𝒫 𝐵) = (𝐵𝑂𝒫 𝐵) | |
16 | 11, 13, 9, 14, 15 | fsovcnvd 38802 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → ◡𝐹 = (𝐵𝑂𝒫 𝐵)) |
17 | 10, 16 | coeq12d 5434 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (◡𝐷 ∘ ◡𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵))) |
18 | 7, 17 | mpdan 705 | . 2 ⊢ (𝜑 → (◡𝐷 ∘ ◡𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵))) |
19 | 4, 18 | syl5eq 2798 | 1 ⊢ (𝜑 → ◡𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1624 ∈ wcel 2131 {crab 3046 Vcvv 3332 ∖ cdif 3704 𝒫 cpw 4294 class class class wbr 4796 ↦ cmpt 4873 ◡ccnv 5257 ∘ ccom 5262 ‘cfv 6041 (class class class)co 6805 ↦ cmpt2 6807 ↑𝑚 cmap 8015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-ral 3047 df-rex 3048 df-reu 3049 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4581 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-id 5166 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-1st 7325 df-2nd 7326 df-map 8017 |
This theorem is referenced by: (None) |
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