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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclsneine0lem | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that at least one (pseudo-)neighborbood of a particular point exists hold equally. (Contributed by RP, 21-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) |
ntrclslem0.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
ntrclsneine0lem | ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾‘𝑠))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6902 | . . . 4 ⊢ (𝑠 = 𝑡 → (𝐼‘𝑠) = (𝐼‘𝑡)) | |
2 | 1 | eleq2d 2815 | . . 3 ⊢ (𝑠 = 𝑡 → (𝑋 ∈ (𝐼‘𝑠) ↔ 𝑋 ∈ (𝐼‘𝑡))) |
3 | 2 | cbvrexvw 3233 | . 2 ⊢ (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑡 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑡)) |
4 | ntrcls.d | . . . . 5 ⊢ 𝐷 = (𝑂‘𝐵) | |
5 | ntrcls.r | . . . . 5 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
6 | 4, 5 | ntrclsrcomplex 43496 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵) |
7 | 6 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵) |
8 | 4, 5 | ntrclsrcomplex 43496 | . . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵) |
9 | 8 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵) |
10 | difeq2 4116 | . . . . . 6 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑡))) | |
11 | 10 | adantl 480 | . . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑡))) |
12 | elpwi 4613 | . . . . . . 7 ⊢ (𝑡 ∈ 𝒫 𝐵 → 𝑡 ⊆ 𝐵) | |
13 | dfss4 4261 | . . . . . . 7 ⊢ (𝑡 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡) | |
14 | 12, 13 | sylib 217 | . . . . . 6 ⊢ (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡) |
15 | 14 | ad2antlr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡) |
16 | 11, 15 | eqtr2d 2769 | . . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → 𝑡 = (𝐵 ∖ 𝑠)) |
17 | 9, 16 | rspcedeq2vd 3619 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵 ∖ 𝑠)) |
18 | fveq2 6902 | . . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝐼‘𝑡) = (𝐼‘(𝐵 ∖ 𝑠))) | |
19 | 18 | eleq2d 2815 | . . . . 5 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝑋 ∈ (𝐼‘𝑡) ↔ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠)))) |
20 | 19 | 3ad2ant3 1132 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑋 ∈ (𝐼‘𝑡) ↔ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠)))) |
21 | ntrcls.o | . . . . . 6 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) | |
22 | 5 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼𝐷𝐾) |
23 | ntrclslem0.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
24 | 23 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵) |
25 | simpr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵) | |
26 | 21, 4, 22, 24, 25 | ntrclselnel2 43519 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠)) ↔ ¬ 𝑋 ∈ (𝐾‘𝑠))) |
27 | 26 | 3adant3 1129 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠)) ↔ ¬ 𝑋 ∈ (𝐾‘𝑠))) |
28 | 20, 27 | bitrd 278 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑋 ∈ (𝐼‘𝑡) ↔ ¬ 𝑋 ∈ (𝐾‘𝑠))) |
29 | 7, 17, 28 | rexxfrd2 5417 | . 2 ⊢ (𝜑 → (∃𝑡 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑡) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾‘𝑠))) |
30 | 3, 29 | bitrid 282 | 1 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾‘𝑠))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3067 Vcvv 3473 ∖ cdif 3946 ⊆ wss 3949 𝒫 cpw 4606 class class class wbr 5152 ↦ cmpt 5235 ‘cfv 6553 (class class class)co 7426 ↑m cmap 8851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-map 8853 |
This theorem is referenced by: ntrclsneine0 43526 |
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