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Mirrors > Home > MPE Home > Th. List > Mathboxes > clsneif1o | 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 operator is a one-to-one, onto mapping. (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 |
---|---|
clsneif1o | ⊢ (𝜑 → 𝐻:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clsnei.d | . . . 4 ⊢ 𝐷 = (𝑃‘𝐵) | |
2 | clsnei.h | . . . 4 ⊢ 𝐻 = (𝐹 ∘ 𝐷) | |
3 | clsnei.r | . . . 4 ⊢ (𝜑 → 𝐾𝐻𝑁) | |
4 | 1, 2, 3 | clsneibex 40805 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
5 | clsnei.o | . . . . 5 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
6 | pwexg 5244 | . . . . . 6 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V) | |
7 | 6 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝒫 𝐵 ∈ V) |
8 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐵 ∈ V) | |
9 | eqid 2798 | . . . . 5 ⊢ (𝒫 𝐵𝑂𝐵) = (𝒫 𝐵𝑂𝐵) | |
10 | 5, 7, 8, 9 | fsovf1od 40717 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (𝒫 𝐵𝑂𝐵):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) |
11 | clsnei.p | . . . . 5 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) | |
12 | eqid 2798 | . . . . 5 ⊢ (𝑃‘𝐵) = (𝑃‘𝐵) | |
13 | 11, 12, 8 | dssmapf1od 40722 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (𝑃‘𝐵):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵)) |
14 | f1oco 6612 | . . . 4 ⊢ (((𝒫 𝐵𝑂𝐵):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ∧ (𝑃‘𝐵):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵)) → ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) | |
15 | 10, 13, 14 | syl2anc 587 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) |
16 | 4, 15 | mpdan 686 | . 2 ⊢ (𝜑 → ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) |
17 | clsnei.f | . . . . 5 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
18 | 17, 1 | coeq12i 5698 | . . . 4 ⊢ (𝐹 ∘ 𝐷) = ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)) |
19 | 2, 18 | eqtri 2821 | . . 3 ⊢ 𝐻 = ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)) |
20 | f1oeq1 6579 | . . 3 ⊢ (𝐻 = ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)) → (𝐻:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ↔ ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))) | |
21 | 19, 20 | ax-mp 5 | . 2 ⊢ (𝐻:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ↔ ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) |
22 | 16, 21 | sylibr 237 | 1 ⊢ (𝜑 → 𝐻:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ∖ cdif 3878 𝒫 cpw 4497 class class class wbr 5030 ↦ cmpt 5110 ∘ ccom 5523 –1-1-onto→wf1o 6323 ‘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: (None) |
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