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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 37420 | . . 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 4776 | . . . . . 6 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V) | |
7 | 6 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝒫 𝐵 ∈ V) |
8 | simpr 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐵 ∈ V) | |
9 | eqid 2610 | . . . . 5 ⊢ (𝒫 𝐵𝑂𝐵) = (𝒫 𝐵𝑂𝐵) | |
10 | 5, 7, 8, 9 | fsovf1od 37330 | . . . 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 2610 | . . . . 5 ⊢ (𝑃‘𝐵) = (𝑃‘𝐵) | |
13 | 11, 12, 8 | dssmapf1od 37335 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (𝑃‘𝐵):(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
14 | f1oco 6072 | . . . 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 691 | . . 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 699 | . 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 5207 | . . . 4 ⊢ (𝐹 ∘ 𝐷) = ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)) |
19 | 2, 18 | eqtri 2632 | . . 3 ⊢ 𝐻 = ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)) |
20 | f1oeq1 6040 | . . 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 223 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ∖ cdif 3537 𝒫 cpw 4108 class class class wbr 4583 ↦ cmpt 4643 ∘ ccom 5042 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 |
This theorem is referenced by: (None) |
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