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Mirrors > Home > MPE Home > Th. List > Mathboxes > dssmapclsntr | Structured version Visualization version GIF version |
Description: The closure and interior operators on a topology are duals of each other. See also kur14lem2 31796. (Contributed by RP, 22-Apr-2021.) |
Ref | Expression |
---|---|
dssmapclsntr.x | ⊢ 𝑋 = ∪ 𝐽 |
dssmapclsntr.k | ⊢ 𝐾 = (cls‘𝐽) |
dssmapclsntr.i | ⊢ 𝐼 = (int‘𝐽) |
dssmapclsntr.o | ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠)))))) |
dssmapclsntr.d | ⊢ 𝐷 = (𝑂‘𝑋) |
Ref | Expression |
---|---|
dssmapclsntr | ⊢ (𝐽 ∈ Top → 𝐾 = (𝐷‘𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dssmapclsntr.x | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
2 | dssmapclsntr.k | . . . . 5 ⊢ 𝐾 = (cls‘𝐽) | |
3 | dssmapclsntr.i | . . . . 5 ⊢ 𝐼 = (int‘𝐽) | |
4 | dssmapclsntr.o | . . . . 5 ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠)))))) | |
5 | dssmapclsntr.d | . . . . 5 ⊢ 𝐷 = (𝑂‘𝑋) | |
6 | 1, 2, 3, 4, 5 | dssmapntrcls 39396 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝐷‘𝐾)) |
7 | 6 | eqcomd 2784 | . . 3 ⊢ (𝐽 ∈ Top → (𝐷‘𝐾) = 𝐼) |
8 | 1 | topopn 21129 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
9 | 4, 5, 8 | dssmapf1od 39285 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐷:(𝒫 𝑋 ↑𝑚 𝒫 𝑋)–1-1-onto→(𝒫 𝑋 ↑𝑚 𝒫 𝑋)) |
10 | 1, 2 | clselmap 39395 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋 ↑𝑚 𝒫 𝑋)) |
11 | f1ocnvfv 6808 | . . . 4 ⊢ ((𝐷:(𝒫 𝑋 ↑𝑚 𝒫 𝑋)–1-1-onto→(𝒫 𝑋 ↑𝑚 𝒫 𝑋) ∧ 𝐾 ∈ (𝒫 𝑋 ↑𝑚 𝒫 𝑋)) → ((𝐷‘𝐾) = 𝐼 → (◡𝐷‘𝐼) = 𝐾)) | |
12 | 9, 10, 11 | syl2anc 579 | . . 3 ⊢ (𝐽 ∈ Top → ((𝐷‘𝐾) = 𝐼 → (◡𝐷‘𝐼) = 𝐾)) |
13 | 7, 12 | mpd 15 | . 2 ⊢ (𝐽 ∈ Top → (◡𝐷‘𝐼) = 𝐾) |
14 | 4, 5, 8 | dssmapnvod 39284 | . . 3 ⊢ (𝐽 ∈ Top → ◡𝐷 = 𝐷) |
15 | 14 | fveq1d 6450 | . 2 ⊢ (𝐽 ∈ Top → (◡𝐷‘𝐼) = (𝐷‘𝐼)) |
16 | 13, 15 | eqtr3d 2816 | 1 ⊢ (𝐽 ∈ Top → 𝐾 = (𝐷‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∖ cdif 3789 𝒫 cpw 4379 ∪ cuni 4673 ↦ cmpt 4967 ◡ccnv 5356 –1-1-onto→wf1o 6136 ‘cfv 6137 (class class class)co 6924 ↑𝑚 cmap 8142 Topctop 21116 intcnt 21240 clsccl 21241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-map 8144 df-top 21117 df-cld 21242 df-ntr 21243 df-cls 21244 |
This theorem is referenced by: (None) |
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