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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 32454. (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 40498 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝐷‘𝐾)) |
7 | 6 | eqcomd 2827 | . . 3 ⊢ (𝐽 ∈ Top → (𝐷‘𝐾) = 𝐼) |
8 | 1 | topopn 21514 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
9 | 4, 5, 8 | dssmapf1od 40387 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋 ↑m 𝒫 𝑋)) |
10 | 1, 2 | clselmap 40497 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋)) |
11 | f1ocnvfv 7035 | . . . 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 586 | . . 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 40386 | . . 3 ⊢ (𝐽 ∈ Top → ◡𝐷 = 𝐷) |
15 | 14 | fveq1d 6672 | . 2 ⊢ (𝐽 ∈ Top → (◡𝐷‘𝐼) = (𝐷‘𝐼)) |
16 | 13, 15 | eqtr3d 2858 | 1 ⊢ (𝐽 ∈ Top → 𝐾 = (𝐷‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∖ cdif 3933 𝒫 cpw 4539 ∪ cuni 4838 ↦ cmpt 5146 ◡ccnv 5554 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 Topctop 21501 intcnt 21625 clsccl 21626 |
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-int 4877 df-iun 4921 df-iin 4922 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 df-top 21502 df-cld 21627 df-ntr 21628 df-cls 21629 |
This theorem is referenced by: (None) |
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