![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > clsneinex | Structured version Visualization version GIF version |
Description: If closure and neighborhoods functions are related, the neighborhoods function exists. (Contributed by RP, 27-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 |
---|---|
clsneinex | ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clsnei.o | . 2 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
2 | clsnei.f | . 2 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
3 | clsnei.d | . . . . 5 ⊢ 𝐷 = (𝑃‘𝐵) | |
4 | clsnei.h | . . . . 5 ⊢ 𝐻 = (𝐹 ∘ 𝐷) | |
5 | clsnei.r | . . . . 5 ⊢ (𝜑 → 𝐾𝐻𝑁) | |
6 | 3, 4, 5 | clsneibex 43155 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
7 | pwexg 5375 | . . . . . . . 8 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V) | |
8 | 7 | adantl 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝒫 𝐵 ∈ V) |
9 | simpr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐵 ∈ V) | |
10 | 1, 8, 9, 2 | fsovf1od 43069 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) |
11 | f1ofn 6833 | . . . . . 6 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → 𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵)) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵)) |
13 | clsnei.p | . . . . . . 7 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) | |
14 | 13, 3, 9 | dssmapf1od 43074 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵)) |
15 | f1of 6832 | . . . . . 6 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)⟶(𝒫 𝐵 ↑m 𝒫 𝐵)) | |
16 | 14, 15 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)⟶(𝒫 𝐵 ↑m 𝒫 𝐵)) |
17 | 5 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐾𝐻𝑁) |
18 | 4 | breqi 5153 | . . . . . 6 ⊢ (𝐾𝐻𝑁 ↔ 𝐾(𝐹 ∘ 𝐷)𝑁) |
19 | 17, 18 | sylib 217 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐾(𝐹 ∘ 𝐷)𝑁) |
20 | 12, 16, 19 | brcoffn 43083 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) |
21 | 6, 20 | mpdan 683 | . . 3 ⊢ (𝜑 → (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) |
22 | 21 | simprd 494 | . 2 ⊢ (𝜑 → (𝐷‘𝐾)𝐹𝑁) |
23 | 1, 2, 22 | ntrneinex 43130 | 1 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 {crab 3430 Vcvv 3472 ∖ cdif 3944 𝒫 cpw 4601 class class class wbr 5147 ↦ cmpt 5230 ∘ ccom 5679 Fn wfn 6537 ⟶wf 6538 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7411 ∈ cmpo 7413 ↑m cmap 8822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-map 8824 |
This theorem is referenced by: clsneifv3 43163 |
Copyright terms: Public domain | W3C validator |