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Mirrors > Home > MPE Home > Th. List > Mathboxes > neicvgnvor | Structured version Visualization version GIF version |
Description: If neighborhood and convergent functions are related by operator 𝐻, the relationship holds with the functions swapped. (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 |
---|---|
neicvgnvor | ⊢ (𝜑 → 𝑀𝐻𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neicvg.r | . . 3 ⊢ (𝜑 → 𝑁𝐻𝑀) | |
2 | neicvg.o | . . . . 5 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
3 | neicvg.p | . . . . 5 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) | |
4 | neicvg.d | . . . . 5 ⊢ 𝐷 = (𝑃‘𝐵) | |
5 | neicvg.f | . . . . 5 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
6 | neicvg.g | . . . . 5 ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) | |
7 | neicvg.h | . . . . 5 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) | |
8 | 2, 3, 4, 5, 6, 7, 1 | neicvgnvo 42292 | . . . 4 ⊢ (𝜑 → ◡𝐻 = 𝐻) |
9 | 8 | breqd 5115 | . . 3 ⊢ (𝜑 → (𝑁◡𝐻𝑀 ↔ 𝑁𝐻𝑀)) |
10 | 1, 9 | mpbird 257 | . 2 ⊢ (𝜑 → 𝑁◡𝐻𝑀) |
11 | relco 6059 | . . . 4 ⊢ Rel (𝐹 ∘ (𝐷 ∘ 𝐺)) | |
12 | 7 | releqi 5732 | . . . 4 ⊢ (Rel 𝐻 ↔ Rel (𝐹 ∘ (𝐷 ∘ 𝐺))) |
13 | 11, 12 | mpbir 230 | . . 3 ⊢ Rel 𝐻 |
14 | 13 | relbrcnv 6058 | . 2 ⊢ (𝑁◡𝐻𝑀 ↔ 𝑀𝐻𝑁) |
15 | 10, 14 | sylib 217 | 1 ⊢ (𝜑 → 𝑀𝐻𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3406 Vcvv 3444 ∖ cdif 3906 𝒫 cpw 4559 class class class wbr 5104 ↦ cmpt 5187 ◡ccnv 5631 ∘ ccom 5636 Rel wrel 5637 ‘cfv 6494 (class class class)co 7352 ∈ cmpo 7354 ↑m cmap 8724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7914 df-2nd 7915 df-map 8726 |
This theorem is referenced by: neicvgnex 42295 |
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