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| Mirrors > Home > MPE Home > Th. List > Mathboxes > clsneiel2 | Structured version Visualization version GIF version | ||
| Description: If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then membership in the closure of the complement of a subset is equivalent to the subset not being a neighborhood of the point. (Contributed by RP, 7-Jun-2021.) |
| Ref | Expression |
|---|---|
| clsnei.o | ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) |
| clsnei.p | ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) |
| clsnei.d | ⊢ 𝐷 = (𝑃‘𝐵) |
| clsnei.f | ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) |
| clsnei.h | ⊢ 𝐻 = (𝐹 ∘ 𝐷) |
| clsnei.r | ⊢ (𝜑 → 𝐾𝐻𝑁) |
| clsneiel.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| clsneiel.s | ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) |
| Ref | Expression |
|---|---|
| clsneiel2 | ⊢ (𝜑 → (𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)) ↔ ¬ 𝑆 ∈ (𝑁‘𝑋))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | clsnei.o | . . 3 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
| 2 | clsnei.p | . . 3 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) | |
| 3 | clsnei.d | . . 3 ⊢ 𝐷 = (𝑃‘𝐵) | |
| 4 | clsnei.f | . . 3 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
| 5 | clsnei.h | . . 3 ⊢ 𝐻 = (𝐹 ∘ 𝐷) | |
| 6 | clsnei.r | . . 3 ⊢ (𝜑 → 𝐾𝐻𝑁) | |
| 7 | clsneiel.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 8 | 3, 5, 6 | clsneircomplex 44093 | . . 3 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
| 9 | 1, 2, 3, 4, 5, 6, 7, 8 | clsneiel1 44098 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)) ↔ ¬ (𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑁‘𝑋))) |
| 10 | clsneiel.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
| 11 | 10 | elpwid 4589 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
| 12 | dfss4 4249 | . . . . 5 ⊢ (𝑆 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑆)) = 𝑆) | |
| 13 | 11, 12 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ 𝑆)) = 𝑆) |
| 14 | 13 | eleq1d 2818 | . . 3 ⊢ (𝜑 → ((𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑁‘𝑋) ↔ 𝑆 ∈ (𝑁‘𝑋))) |
| 15 | 14 | notbid 318 | . 2 ⊢ (𝜑 → (¬ (𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑁‘𝑋) ↔ ¬ 𝑆 ∈ (𝑁‘𝑋))) |
| 16 | 9, 15 | bitrd 279 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)) ↔ ¬ 𝑆 ∈ (𝑁‘𝑋))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 {crab 3419 Vcvv 3463 ∖ cdif 3928 ⊆ wss 3931 𝒫 cpw 4580 class class class wbr 5123 ↦ cmpt 5205 ∘ ccom 5669 ‘cfv 6541 (class class class)co 7413 ∈ cmpo 7415 ↑m cmap 8848 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7996 df-2nd 7997 df-map 8850 |
| This theorem is referenced by: clsneifv3 44100 |
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