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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 6448 | . . . . 5 ⊢ (𝑚 = 𝑆 → (𝐼‘𝑚) = (𝐼‘𝑆)) | |
3 | 2 | eleq2d 2845 | . . . 4 ⊢ (𝑚 = 𝑆 → (𝑋 ∈ (𝐼‘𝑚) ↔ 𝑋 ∈ (𝐼‘𝑆))) |
4 | 3 | elrab3 3574 | . . 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 39337 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
10 | 9 | pwexd 5093 | . . . . 5 ⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
11 | 6, 7, 8 | ntrneiiex 39340 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
12 | eqid 2778 | . . . . 5 ⊢ (𝐹‘𝐼) = (𝐹‘𝐼) | |
13 | ntrnei.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
14 | 6, 10, 9, 7, 11, 12, 13 | fsovfvfvd 39271 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝐼)‘𝑋) = {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)}) |
15 | 6, 7, 8 | ntrneifv1 39343 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐼) = 𝑁) |
16 | 15 | fveq1d 6450 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝐼)‘𝑋) = (𝑁‘𝑋)) |
17 | 14, 16 | eqtr3d 2816 | . . 3 ⊢ (𝜑 → {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} = (𝑁‘𝑋)) |
18 | 17 | eleq2d 2845 | . 2 ⊢ (𝜑 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} ↔ 𝑆 ∈ (𝑁‘𝑋))) |
19 | 5, 18 | bitr3d 273 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑆 ∈ (𝑁‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 {crab 3094 Vcvv 3398 𝒫 cpw 4379 class class class wbr 4888 ↦ cmpt 4967 ‘cfv 6137 (class class class)co 6924 ↦ cmpt2 6926 ↑𝑚 cmap 8142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-map 8144 |
This theorem is referenced by: ntrneifv3 39346 ntrneineine0lem 39347 ntrneineine1lem 39348 ntrneifv4 39349 ntrneiel2 39350 ntrneicls00 39353 ntrneicls11 39354 ntrneiiso 39355 ntrneik2 39356 ntrneix2 39357 ntrneikb 39358 ntrneixb 39359 ntrneik3 39360 ntrneix3 39361 ntrneik13 39362 ntrneix13 39363 ntrneik4w 39364 ntrneik4 39365 clsneiel1 39372 neicvgel1 39383 |
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