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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > neicvgf1o | Structured version Visualization version GIF version |
Description: If neighborhood and convergent functions are related by operator 𝐻, it is a one-to-one onto relation. (Contributed by RP, 11-Jun-2021.) |
Ref | Expression |
---|---|
neicvg.o | ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) |
neicvg.p | ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) |
neicvg.d | ⊢ 𝐷 = (𝑃‘𝐵) |
neicvg.f | ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) |
neicvg.g | ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) |
neicvg.h | ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) |
neicvg.r | ⊢ (𝜑 → 𝑁𝐻𝑀) |
Ref | Expression |
---|---|
neicvgf1o | ⊢ (𝜑 → 𝐻:(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neicvg.o | . . . 4 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
2 | neicvg.d | . . . . . 6 ⊢ 𝐷 = (𝑃‘𝐵) | |
3 | neicvg.h | . . . . . 6 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) | |
4 | neicvg.r | . . . . . 6 ⊢ (𝜑 → 𝑁𝐻𝑀) | |
5 | 2, 3, 4 | neicvgbex 39831 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
6 | 5 | pwexd 5133 | . . . 4 ⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
7 | neicvg.f | . . . 4 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
8 | 1, 6, 5, 7 | fsovf1od 39731 | . . 3 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵)) |
9 | neicvg.p | . . . . 5 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) | |
10 | 9, 2, 5 | dssmapf1od 39736 | . . . 4 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
11 | neicvg.g | . . . . 5 ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) | |
12 | 1, 5, 6, 11 | fsovf1od 39731 | . . . 4 ⊢ (𝜑 → 𝐺:(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
13 | f1oco 6466 | . . . 4 ⊢ ((𝐷:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ 𝐺:(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (𝐷 ∘ 𝐺):(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵)) | |
14 | 10, 12, 13 | syl2anc 576 | . . 3 ⊢ (𝜑 → (𝐷 ∘ 𝐺):(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
15 | f1oco 6466 | . . 3 ⊢ ((𝐹:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵) ∧ (𝐷 ∘ 𝐺):(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (𝐹 ∘ (𝐷 ∘ 𝐺)):(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵)) | |
16 | 8, 14, 15 | syl2anc 576 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝐷 ∘ 𝐺)):(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵)) |
17 | f1oeq1 6433 | . . 3 ⊢ (𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) → (𝐻:(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵) ↔ (𝐹 ∘ (𝐷 ∘ 𝐺)):(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵))) | |
18 | 3, 17 | ax-mp 5 | . 2 ⊢ (𝐻:(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵) ↔ (𝐹 ∘ (𝐷 ∘ 𝐺)):(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵)) |
19 | 16, 18 | sylibr 226 | 1 ⊢ (𝜑 → 𝐻:(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2050 {crab 3092 Vcvv 3415 ∖ cdif 3826 𝒫 cpw 4422 class class class wbr 4929 ↦ cmpt 5008 ∘ ccom 5411 –1-1-onto→wf1o 6187 ‘cfv 6188 (class class class)co 6976 ∈ cmpo 6978 ↑𝑚 cmap 8206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-1st 7501 df-2nd 7502 df-map 8208 |
This theorem is referenced by: (None) |
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