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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclselnel1 | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is an equivalence between membership in the interior of a set and non-membership in the closure of the complement of the set. (Contributed by RP, 28-May-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) |
ntrcls.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ntrcls.s | ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) |
Ref | Expression |
---|---|
ntrclselnel1 | ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrcls.o | . . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) | |
2 | ntrcls.d | . . . . . . 7 ⊢ 𝐷 = (𝑂‘𝐵) | |
3 | ntrcls.r | . . . . . . 7 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
4 | 1, 2, 3 | ntrclsfv2 37374 | . . . . . 6 ⊢ (𝜑 → (𝐷‘𝐾) = 𝐼) |
5 | 4 | eqcomd 2616 | . . . . 5 ⊢ (𝜑 → 𝐼 = (𝐷‘𝐾)) |
6 | 5 | fveq1d 6105 | . . . 4 ⊢ (𝜑 → (𝐼‘𝑆) = ((𝐷‘𝐾)‘𝑆)) |
7 | 2, 3 | ntrclsbex 37352 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
8 | 1, 2, 3 | ntrclskex 37372 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
9 | eqid 2610 | . . . . 5 ⊢ (𝐷‘𝐾) = (𝐷‘𝐾) | |
10 | ntrcls.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
11 | eqid 2610 | . . . . 5 ⊢ ((𝐷‘𝐾)‘𝑆) = ((𝐷‘𝐾)‘𝑆) | |
12 | 1, 2, 7, 8, 9, 10, 11 | dssmapfv3d 37333 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐾)‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))) |
13 | 6, 12 | eqtrd 2644 | . . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))) |
14 | 13 | eleq2d 2673 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))) |
15 | ntrcls.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
16 | eldif 3550 | . . . 4 ⊢ (𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)))) | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆))))) |
18 | 15, 17 | mpbirand 529 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)))) |
19 | 14, 18 | bitrd 267 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 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: ntrclselnel2 37376 clsneiel1 37426 neicvgel1 37437 |
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