Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > clsneicnv | 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 converse of the operator is known. (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 |
---|---|
clsneicnv | ⊢ (𝜑 → ◡𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clsnei.h | . . . 4 ⊢ 𝐻 = (𝐹 ∘ 𝐷) | |
2 | 1 | cnveqi 5745 | . . 3 ⊢ ◡𝐻 = ◡(𝐹 ∘ 𝐷) |
3 | cnvco 5756 | . . 3 ⊢ ◡(𝐹 ∘ 𝐷) = (◡𝐷 ∘ ◡𝐹) | |
4 | 2, 3 | eqtri 2844 | . 2 ⊢ ◡𝐻 = (◡𝐷 ∘ ◡𝐹) |
5 | clsnei.d | . . . 4 ⊢ 𝐷 = (𝑃‘𝐵) | |
6 | clsnei.r | . . . 4 ⊢ (𝜑 → 𝐾𝐻𝑁) | |
7 | 5, 1, 6 | clsneibex 40472 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
8 | clsnei.p | . . . . 5 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) | |
9 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐵 ∈ V) | |
10 | 8, 5, 9 | dssmapnvod 40386 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → ◡𝐷 = 𝐷) |
11 | clsnei.o | . . . . 5 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
12 | pwexg 5279 | . . . . . 6 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V) | |
13 | 12 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝒫 𝐵 ∈ V) |
14 | clsnei.f | . . . . 5 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
15 | eqid 2821 | . . . . 5 ⊢ (𝐵𝑂𝒫 𝐵) = (𝐵𝑂𝒫 𝐵) | |
16 | 11, 13, 9, 14, 15 | fsovcnvd 40380 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → ◡𝐹 = (𝐵𝑂𝒫 𝐵)) |
17 | 10, 16 | coeq12d 5735 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (◡𝐷 ∘ ◡𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵))) |
18 | 7, 17 | mpdan 685 | . 2 ⊢ (𝜑 → (◡𝐷 ∘ ◡𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵))) |
19 | 4, 18 | syl5eq 2868 | 1 ⊢ (𝜑 → ◡𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3142 Vcvv 3494 ∖ cdif 3933 𝒫 cpw 4539 class class class wbr 5066 ↦ cmpt 5146 ◡ccnv 5554 ∘ ccom 5559 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ↑m cmap 8406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |