![]() |
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 43156 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
7 | pwexg 5376 | . . . . . . . 8 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V) | |
8 | 7 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝒫 𝐵 ∈ V) |
9 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐵 ∈ V) | |
10 | 1, 8, 9, 2 | fsovf1od 43070 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) |
11 | f1ofn 6834 | . . . . . 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 43075 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵)) |
15 | f1of 6833 | . . . . . 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 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐾𝐻𝑁) |
18 | 4 | breqi 5154 | . . . . . 6 ⊢ (𝐾𝐻𝑁 ↔ 𝐾(𝐹 ∘ 𝐷)𝑁) |
19 | 17, 18 | sylib 217 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐾(𝐹 ∘ 𝐷)𝑁) |
20 | 12, 16, 19 | brcoffn 43084 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) |
21 | 6, 20 | mpdan 684 | . . 3 ⊢ (𝜑 → (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) |
22 | 21 | simprd 495 | . 2 ⊢ (𝜑 → (𝐷‘𝐾)𝐹𝑁) |
23 | 1, 2, 22 | ntrneinex 43131 | 1 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {crab 3431 Vcvv 3473 ∖ cdif 3945 𝒫 cpw 4602 class class class wbr 5148 ↦ cmpt 5231 ∘ ccom 5680 Fn wfn 6538 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 ↑m cmap 8824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-map 8826 |
This theorem is referenced by: clsneifv3 43164 |
Copyright terms: Public domain | W3C validator |