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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrneiel | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, 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, 29-May-2021.) |
Ref | Expression |
---|---|
ntrnei.o | ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) |
ntrnei.f | ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) |
ntrnei.r | ⊢ (𝜑 → 𝐼𝐹𝑁) |
ntrnei.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ntrnei.s | ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) |
Ref | Expression |
---|---|
ntrneiel | ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑆 ∈ (𝑁‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrnei.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
2 | fveq2 6891 | . . . . 5 ⊢ (𝑚 = 𝑆 → (𝐼‘𝑚) = (𝐼‘𝑆)) | |
3 | 2 | eleq2d 2812 | . . . 4 ⊢ (𝑚 = 𝑆 → (𝑋 ∈ (𝐼‘𝑚) ↔ 𝑋 ∈ (𝐼‘𝑆))) |
4 | 3 | elrab3 3682 | . . 3 ⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} ↔ 𝑋 ∈ (𝐼‘𝑆))) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝜑 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} ↔ 𝑋 ∈ (𝐼‘𝑆))) |
6 | ntrnei.o | . . . . 5 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
7 | ntrnei.f | . . . . . . 7 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
8 | ntrnei.r | . . . . . . 7 ⊢ (𝜑 → 𝐼𝐹𝑁) | |
9 | 6, 7, 8 | ntrneibex 43775 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
10 | 9 | pwexd 5374 | . . . . 5 ⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
11 | 6, 7, 8 | ntrneiiex 43778 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
12 | eqid 2726 | . . . . 5 ⊢ (𝐹‘𝐼) = (𝐹‘𝐼) | |
13 | ntrnei.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
14 | 6, 10, 9, 7, 11, 12, 13 | fsovfvfvd 43713 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝐼)‘𝑋) = {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)}) |
15 | 6, 7, 8 | ntrneifv1 43781 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐼) = 𝑁) |
16 | 15 | fveq1d 6893 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝐼)‘𝑋) = (𝑁‘𝑋)) |
17 | 14, 16 | eqtr3d 2768 | . . 3 ⊢ (𝜑 → {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} = (𝑁‘𝑋)) |
18 | 17 | eleq2d 2812 | . 2 ⊢ (𝜑 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} ↔ 𝑆 ∈ (𝑁‘𝑋))) |
19 | 5, 18 | bitr3d 280 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑆 ∈ (𝑁‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 {crab 3420 Vcvv 3463 𝒫 cpw 4598 class class class wbr 5144 ↦ cmpt 5227 ‘cfv 6544 (class class class)co 7414 ∈ cmpo 7416 ↑m cmap 8845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7993 df-2nd 7994 df-map 8847 |
This theorem is referenced by: ntrneifv3 43784 ntrneineine0lem 43785 ntrneineine1lem 43786 ntrneifv4 43787 ntrneiel2 43788 ntrneicls00 43791 ntrneicls11 43792 ntrneiiso 43793 ntrneik2 43794 ntrneix2 43795 ntrneikb 43796 ntrneixb 43797 ntrneik3 43798 ntrneix3 43799 ntrneik13 43800 ntrneix13 43801 ntrneik4w 43802 ntrneik4 43803 clsneiel1 43810 neicvgel1 43821 |
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