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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 32351. (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 40356 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝐷‘𝐾)) |
7 | 6 | eqcomd 2824 | . . 3 ⊢ (𝐽 ∈ Top → (𝐷‘𝐾) = 𝐼) |
8 | 1 | topopn 21442 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
9 | 4, 5, 8 | dssmapf1od 40245 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋 ↑m 𝒫 𝑋)) |
10 | 1, 2 | clselmap 40355 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋)) |
11 | f1ocnvfv 7026 | . . . 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 40244 | . . 3 ⊢ (𝐽 ∈ Top → ◡𝐷 = 𝐷) |
15 | 14 | fveq1d 6665 | . 2 ⊢ (𝐽 ∈ Top → (◡𝐷‘𝐼) = (𝐷‘𝐼)) |
16 | 13, 15 | eqtr3d 2855 | 1 ⊢ (𝐽 ∈ Top → 𝐾 = (𝐷‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 𝒫 cpw 4535 ∪ cuni 4830 ↦ cmpt 5137 ◡ccnv 5547 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 Topctop 21429 intcnt 21553 clsccl 21554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-map 8397 df-top 21430 df-cld 21555 df-ntr 21556 df-cls 21557 |
This theorem is referenced by: (None) |
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