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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclscls00 | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.) |
Ref | Expression |
---|---|
ntrcls.o | ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) |
ntrcls.d | ⊢ 𝐷 = (𝑂‘𝐵) |
ntrcls.r | ⊢ (𝜑 → 𝐼𝐷𝐾) |
Ref | Expression |
---|---|
ntrclscls00 | ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrcls.o | . . . . . 6 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) | |
2 | ntrcls.d | . . . . . 6 ⊢ 𝐷 = (𝑂‘𝐵) | |
3 | ntrcls.r | . . . . . 6 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
4 | 1, 2, 3 | ntrclsfv1 37373 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾) |
5 | 4 | fveq1d 6105 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐼)‘∅) = (𝐾‘∅)) |
6 | 2, 3 | ntrclsbex 37352 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
7 | 1, 2, 3 | ntrclsiex 37371 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
8 | eqid 2610 | . . . . 5 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼) | |
9 | 0elpw 4760 | . . . . . 6 ⊢ ∅ ∈ 𝒫 𝐵 | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝜑 → ∅ ∈ 𝒫 𝐵) |
11 | eqid 2610 | . . . . 5 ⊢ ((𝐷‘𝐼)‘∅) = ((𝐷‘𝐼)‘∅) | |
12 | 1, 2, 6, 7, 8, 10, 11 | dssmapfv3d 37333 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐼)‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅)))) |
13 | 5, 12 | eqtr3d 2646 | . . 3 ⊢ (𝜑 → (𝐾‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅)))) |
14 | dif0 3904 | . . . . . . 7 ⊢ (𝐵 ∖ ∅) = 𝐵 | |
15 | 14 | fveq2i 6106 | . . . . . 6 ⊢ (𝐼‘(𝐵 ∖ ∅)) = (𝐼‘𝐵) |
16 | id 22 | . . . . . 6 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐼‘𝐵) = 𝐵) | |
17 | 15, 16 | syl5eq 2656 | . . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐼‘(𝐵 ∖ ∅)) = 𝐵) |
18 | 17 | difeq2d 3690 | . . . 4 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = (𝐵 ∖ 𝐵)) |
19 | difid 3902 | . . . 4 ⊢ (𝐵 ∖ 𝐵) = ∅ | |
20 | 18, 19 | syl6eq 2660 | . . 3 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = ∅) |
21 | 13, 20 | sylan9eq 2664 | . 2 ⊢ ((𝜑 ∧ (𝐼‘𝐵) = 𝐵) → (𝐾‘∅) = ∅) |
22 | pwidg 4121 | . . . . 5 ⊢ (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵) | |
23 | 6, 22 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵) |
24 | 1, 2, 3, 23 | ntrclsfv 37377 | . . 3 ⊢ (𝜑 → (𝐼‘𝐵) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵)))) |
25 | 19 | fveq2i 6106 | . . . . . 6 ⊢ (𝐾‘(𝐵 ∖ 𝐵)) = (𝐾‘∅) |
26 | id 22 | . . . . . 6 ⊢ ((𝐾‘∅) = ∅ → (𝐾‘∅) = ∅) | |
27 | 25, 26 | syl5eq 2656 | . . . . 5 ⊢ ((𝐾‘∅) = ∅ → (𝐾‘(𝐵 ∖ 𝐵)) = ∅) |
28 | 27 | difeq2d 3690 | . . . 4 ⊢ ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))) = (𝐵 ∖ ∅)) |
29 | 28, 14 | syl6eq 2660 | . . 3 ⊢ ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))) = 𝐵) |
30 | 24, 29 | sylan9eq 2664 | . 2 ⊢ ((𝜑 ∧ (𝐾‘∅) = ∅) → (𝐼‘𝐵) = 𝐵) |
31 | 21, 30 | impbida 873 | 1 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 𝒫 cpw 4108 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 |
This theorem is referenced by: (None) |
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