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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 31189. (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 38426 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝐷‘𝐾)) |
7 | 6 | eqcomd 2628 | . . 3 ⊢ (𝐽 ∈ Top → (𝐷‘𝐾) = 𝐼) |
8 | 1 | topopn 20711 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
9 | 4, 5, 8 | dssmapf1od 38315 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐷:(𝒫 𝑋 ↑𝑚 𝒫 𝑋)–1-1-onto→(𝒫 𝑋 ↑𝑚 𝒫 𝑋)) |
10 | 1, 2 | clselmap 38425 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋 ↑𝑚 𝒫 𝑋)) |
11 | f1ocnvfv 6534 | . . . 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 693 | . . 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 38314 | . . 3 ⊢ (𝐽 ∈ Top → ◡𝐷 = 𝐷) |
15 | 14 | fveq1d 6193 | . 2 ⊢ (𝐽 ∈ Top → (◡𝐷‘𝐼) = (𝐷‘𝐼)) |
16 | 13, 15 | eqtr3d 2658 | 1 ⊢ (𝐽 ∈ Top → 𝐾 = (𝐷‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 𝒫 cpw 4158 ∪ cuni 4436 ↦ cmpt 4729 ◡ccnv 5113 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 Topctop 20698 intcnt 20821 clsccl 20822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-top 20699 df-cld 20823 df-ntr 20824 df-cls 20825 |
This theorem is referenced by: (None) |
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