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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 34186. (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 42864 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝐷‘𝐾)) |
7 | 6 | eqcomd 2738 | . . 3 ⊢ (𝐽 ∈ Top → (𝐷‘𝐾) = 𝐼) |
8 | 1 | topopn 22399 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
9 | 4, 5, 8 | dssmapf1od 42757 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋 ↑m 𝒫 𝑋)) |
10 | 1, 2 | clselmap 42863 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋)) |
11 | f1ocnvfv 7272 | . . . 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 584 | . . 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 42756 | . . 3 ⊢ (𝐽 ∈ Top → ◡𝐷 = 𝐷) |
15 | 14 | fveq1d 6890 | . 2 ⊢ (𝐽 ∈ Top → (◡𝐷‘𝐼) = (𝐷‘𝐼)) |
16 | 13, 15 | eqtr3d 2774 | 1 ⊢ (𝐽 ∈ Top → 𝐾 = (𝐷‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∖ cdif 3944 𝒫 cpw 4601 ∪ cuni 4907 ↦ cmpt 5230 ◡ccnv 5674 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7405 ↑m cmap 8816 Topctop 22386 intcnt 22512 clsccl 22513 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-map 8818 df-top 22387 df-cld 22514 df-ntr 22515 df-cls 22516 |
This theorem is referenced by: (None) |
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