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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 6433 | . . . 4 ⊢ (𝑠 = 𝑡 → (𝐼‘𝑠) = (𝐼‘𝑡)) | |
2 | 1 | eleq2d 2892 | . . 3 ⊢ (𝑠 = 𝑡 → (𝑋 ∈ (𝐼‘𝑠) ↔ 𝑋 ∈ (𝐼‘𝑡))) |
3 | 2 | cbvrexv 3384 | . 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 39166 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵) |
7 | 6 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵) |
8 | 4, 5 | ntrclsrcomplex 39166 | . . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵) |
9 | 8 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵) |
10 | difeq2 3949 | . . . . . 6 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑡))) | |
11 | 10 | adantl 475 | . . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑡))) |
12 | elpwi 4388 | . . . . . . 7 ⊢ (𝑡 ∈ 𝒫 𝐵 → 𝑡 ⊆ 𝐵) | |
13 | dfss4 4088 | . . . . . . 7 ⊢ (𝑡 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡) | |
14 | 12, 13 | sylib 210 | . . . . . 6 ⊢ (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡) |
15 | 14 | ad2antlr 718 | . . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡) |
16 | 11, 15 | eqtr2d 2862 | . . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → 𝑡 = (𝐵 ∖ 𝑠)) |
17 | 9, 16 | rspcedeq2vd 3536 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵 ∖ 𝑠)) |
18 | fveq2 6433 | . . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝐼‘𝑡) = (𝐼‘(𝐵 ∖ 𝑠))) | |
19 | 18 | eleq2d 2892 | . . . . 5 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝑋 ∈ (𝐼‘𝑡) ↔ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠)))) |
20 | 19 | 3ad2ant3 1169 | . . . 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 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼𝐷𝐾) |
23 | ntrclslem0.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
24 | 23 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵) |
25 | simpr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵) | |
26 | 21, 4, 22, 24, 25 | ntrclselnel2 39189 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠)) ↔ ¬ 𝑋 ∈ (𝐾‘𝑠))) |
27 | 26 | 3adant3 1166 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠)) ↔ ¬ 𝑋 ∈ (𝐾‘𝑠))) |
28 | 20, 27 | bitrd 271 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑋 ∈ (𝐼‘𝑡) ↔ ¬ 𝑋 ∈ (𝐾‘𝑠))) |
29 | 7, 17, 28 | rexxfrd2 5113 | . 2 ⊢ (𝜑 → (∃𝑡 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑡) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾‘𝑠))) |
30 | 3, 29 | syl5bb 275 | 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 198 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 Vcvv 3414 ∖ cdif 3795 ⊆ wss 3798 𝒫 cpw 4378 class class class wbr 4873 ↦ cmpt 4952 ‘cfv 6123 (class class class)co 6905 ↑𝑚 cmap 8122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-map 8124 |
This theorem is referenced by: ntrclsneine0 39196 |
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