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Theorem ntrclsneine0lem 44082
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.)
Hypotheses
Ref Expression
ntrcls.o 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
ntrcls.d 𝐷 = (𝑂𝐵)
ntrcls.r (𝜑𝐼𝐷𝐾)
ntrclslem0.x (𝜑𝑋𝐵)
Assertion
Ref Expression
ntrclsneine0lem (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾𝑠)))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑠   𝑗,𝐼,𝑘,𝑠   𝑋,𝑠   𝜑,𝑖,𝑗,𝑘,𝑠
Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘,𝑠)   𝐼(𝑖)   𝐾(𝑖,𝑗,𝑘,𝑠)   𝑂(𝑖,𝑗,𝑘,𝑠)   𝑋(𝑖,𝑗,𝑘)

Proof of Theorem ntrclsneine0lem
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6905 . . . 4 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
21eleq2d 2826 . . 3 (𝑠 = 𝑡 → (𝑋 ∈ (𝐼𝑠) ↔ 𝑋 ∈ (𝐼𝑡)))
32cbvrexvw 3237 . 2 (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ ∃𝑡 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑡))
4 ntrcls.d . . . . 5 𝐷 = (𝑂𝐵)
5 ntrcls.r . . . . 5 (𝜑𝐼𝐷𝐾)
64, 5ntrclsrcomplex 44053 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
76adantr 480 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
84, 5ntrclsrcomplex 44053 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
98adantr 480 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
10 difeq2 4119 . . . . . 6 (𝑠 = (𝐵𝑡) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
1110adantl 481 . . . . 5 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
12 elpwi 4606 . . . . . . 7 (𝑡 ∈ 𝒫 𝐵𝑡𝐵)
13 dfss4 4268 . . . . . . 7 (𝑡𝐵 ↔ (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1412, 13sylib 218 . . . . . 6 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1514ad2antlr 727 . . . . 5 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1611, 15eqtr2d 2777 . . . 4 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → 𝑡 = (𝐵𝑠))
179, 16rspcedeq2vd 3629 . . 3 ((𝜑𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠))
18 fveq2 6905 . . . . . 6 (𝑡 = (𝐵𝑠) → (𝐼𝑡) = (𝐼‘(𝐵𝑠)))
1918eleq2d 2826 . . . . 5 (𝑡 = (𝐵𝑠) → (𝑋 ∈ (𝐼𝑡) ↔ 𝑋 ∈ (𝐼‘(𝐵𝑠))))
20193ad2ant3 1135 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝑋 ∈ (𝐼𝑡) ↔ 𝑋 ∈ (𝐼‘(𝐵𝑠))))
21 ntrcls.o . . . . . 6 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
225adantr 480 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐼𝐷𝐾)
23 ntrclslem0.x . . . . . . 7 (𝜑𝑋𝐵)
2423adantr 480 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
25 simpr 484 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
2621, 4, 22, 24, 25ntrclselnel2 44076 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼‘(𝐵𝑠)) ↔ ¬ 𝑋 ∈ (𝐾𝑠)))
27263adant3 1132 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝑋 ∈ (𝐼‘(𝐵𝑠)) ↔ ¬ 𝑋 ∈ (𝐾𝑠)))
2820, 27bitrd 279 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝑋 ∈ (𝐼𝑡) ↔ ¬ 𝑋 ∈ (𝐾𝑠)))
297, 17, 28rexxfrd2 5412 . 2 (𝜑 → (∃𝑡 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑡) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾𝑠)))
303, 29bitrid 283 1 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾𝑠)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wrex 3069  Vcvv 3479  cdif 3947  wss 3950  𝒫 cpw 4599   class class class wbr 5142  cmpt 5224  cfv 6560  (class class class)co 7432  m cmap 8867
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 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
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 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869
This theorem is referenced by:  ntrclsneine0  44083
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