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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 6695 | . . . . 5 ⊢ (𝑚 = 𝑆 → (𝐼‘𝑚) = (𝐼‘𝑆)) | |
3 | 2 | eleq2d 2816 | . . . 4 ⊢ (𝑚 = 𝑆 → (𝑋 ∈ (𝐼‘𝑚) ↔ 𝑋 ∈ (𝐼‘𝑆))) |
4 | 3 | elrab3 3592 | . . 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 41301 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
10 | 9 | pwexd 5257 | . . . . 5 ⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
11 | 6, 7, 8 | ntrneiiex 41304 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
12 | eqid 2736 | . . . . 5 ⊢ (𝐹‘𝐼) = (𝐹‘𝐼) | |
13 | ntrnei.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
14 | 6, 10, 9, 7, 11, 12, 13 | fsovfvfvd 41237 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝐼)‘𝑋) = {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)}) |
15 | 6, 7, 8 | ntrneifv1 41307 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐼) = 𝑁) |
16 | 15 | fveq1d 6697 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝐼)‘𝑋) = (𝑁‘𝑋)) |
17 | 14, 16 | eqtr3d 2773 | . . 3 ⊢ (𝜑 → {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} = (𝑁‘𝑋)) |
18 | 17 | eleq2d 2816 | . 2 ⊢ (𝜑 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} ↔ 𝑆 ∈ (𝑁‘𝑋))) |
19 | 5, 18 | bitr3d 284 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑆 ∈ (𝑁‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 {crab 3055 Vcvv 3398 𝒫 cpw 4499 class class class wbr 5039 ↦ cmpt 5120 ‘cfv 6358 (class class class)co 7191 ∈ cmpo 7193 ↑m cmap 8486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-map 8488 |
This theorem is referenced by: ntrneifv3 41310 ntrneineine0lem 41311 ntrneineine1lem 41312 ntrneifv4 41313 ntrneiel2 41314 ntrneicls00 41317 ntrneicls11 41318 ntrneiiso 41319 ntrneik2 41320 ntrneix2 41321 ntrneikb 41322 ntrneixb 41323 ntrneik3 41324 ntrneix3 41325 ntrneik13 41326 ntrneix13 41327 ntrneik4w 41328 ntrneik4 41329 clsneiel1 41336 neicvgel1 41347 |
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