| Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > neicvgnvo | Structured version Visualization version GIF version | ||
| Description: If neighborhood and convergent functions are related by operator 𝐻, it is its own converse function. (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 |
|---|---|
| neicvgnvo | ⊢ (𝜑 → ◡𝐻 = 𝐻) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | neicvg.h | . . . . 5 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) | |
| 2 | 1 | cnveqi 5885 | . . . 4 ⊢ ◡𝐻 = ◡(𝐹 ∘ (𝐷 ∘ 𝐺)) |
| 3 | cnvco 5896 | . . . 4 ⊢ ◡(𝐹 ∘ (𝐷 ∘ 𝐺)) = (◡(𝐷 ∘ 𝐺) ∘ ◡𝐹) | |
| 4 | cnvco 5896 | . . . . 5 ⊢ ◡(𝐷 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐷) | |
| 5 | 4 | coeq1i 5870 | . . . 4 ⊢ (◡(𝐷 ∘ 𝐺) ∘ ◡𝐹) = ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹) |
| 6 | 2, 3, 5 | 3eqtri 2769 | . . 3 ⊢ ◡𝐻 = ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹) |
| 7 | neicvg.o | . . . . . 6 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
| 8 | neicvg.d | . . . . . . 7 ⊢ 𝐷 = (𝑃‘𝐵) | |
| 9 | neicvg.r | . . . . . . 7 ⊢ (𝜑 → 𝑁𝐻𝑀) | |
| 10 | 8, 1, 9 | neicvgbex 44125 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
| 11 | 10 | pwexd 5379 | . . . . . 6 ⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
| 12 | neicvg.g | . . . . . 6 ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) | |
| 13 | neicvg.f | . . . . . 6 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
| 14 | 7, 10, 11, 12, 13 | fsovcnvd 44027 | . . . . 5 ⊢ (𝜑 → ◡𝐺 = 𝐹) |
| 15 | neicvg.p | . . . . . 6 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) | |
| 16 | 15, 8, 10 | dssmapnvod 44033 | . . . . 5 ⊢ (𝜑 → ◡𝐷 = 𝐷) |
| 17 | 14, 16 | coeq12d 5875 | . . . 4 ⊢ (𝜑 → (◡𝐺 ∘ ◡𝐷) = (𝐹 ∘ 𝐷)) |
| 18 | 7, 11, 10, 13, 12 | fsovcnvd 44027 | . . . 4 ⊢ (𝜑 → ◡𝐹 = 𝐺) |
| 19 | 17, 18 | coeq12d 5875 | . . 3 ⊢ (𝜑 → ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹) = ((𝐹 ∘ 𝐷) ∘ 𝐺)) |
| 20 | 6, 19 | eqtrid 2789 | . 2 ⊢ (𝜑 → ◡𝐻 = ((𝐹 ∘ 𝐷) ∘ 𝐺)) |
| 21 | coass 6285 | . . 3 ⊢ ((𝐹 ∘ 𝐷) ∘ 𝐺) = (𝐹 ∘ (𝐷 ∘ 𝐺)) | |
| 22 | 21, 1 | eqtr4i 2768 | . 2 ⊢ ((𝐹 ∘ 𝐷) ∘ 𝐺) = 𝐻 |
| 23 | 20, 22 | eqtrdi 2793 | 1 ⊢ (𝜑 → ◡𝐻 = 𝐻) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ∖ cdif 3948 𝒫 cpw 4600 class class class wbr 5143 ↦ cmpt 5225 ◡ccnv 5684 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ↑m cmap 8866 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 |
| This theorem is referenced by: neicvgnvor 44129 |
| Copyright terms: Public domain | W3C validator |