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Theorem ntrclsneine0lem 43525
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 6902 . . . 4 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
21eleq2d 2815 . . 3 (𝑠 = 𝑡 → (𝑋 ∈ (𝐼𝑠) ↔ 𝑋 ∈ (𝐼𝑡)))
32cbvrexvw 3233 . 2 (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ ∃𝑡 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑡))
4 ntrcls.d . . . . 5 𝐷 = (𝑂𝐵)
5 ntrcls.r . . . . 5 (𝜑𝐼𝐷𝐾)
64, 5ntrclsrcomplex 43496 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
76adantr 479 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
84, 5ntrclsrcomplex 43496 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
98adantr 479 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
10 difeq2 4116 . . . . . 6 (𝑠 = (𝐵𝑡) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
1110adantl 480 . . . . 5 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
12 elpwi 4613 . . . . . . 7 (𝑡 ∈ 𝒫 𝐵𝑡𝐵)
13 dfss4 4261 . . . . . . 7 (𝑡𝐵 ↔ (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1412, 13sylib 217 . . . . . 6 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1514ad2antlr 725 . . . . 5 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1611, 15eqtr2d 2769 . . . 4 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → 𝑡 = (𝐵𝑠))
179, 16rspcedeq2vd 3619 . . 3 ((𝜑𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠))
18 fveq2 6902 . . . . . 6 (𝑡 = (𝐵𝑠) → (𝐼𝑡) = (𝐼‘(𝐵𝑠)))
1918eleq2d 2815 . . . . 5 (𝑡 = (𝐵𝑠) → (𝑋 ∈ (𝐼𝑡) ↔ 𝑋 ∈ (𝐼‘(𝐵𝑠))))
20193ad2ant3 1132 . . . 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 479 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐼𝐷𝐾)
23 ntrclslem0.x . . . . . . 7 (𝜑𝑋𝐵)
2423adantr 479 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
25 simpr 483 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
2621, 4, 22, 24, 25ntrclselnel2 43519 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼‘(𝐵𝑠)) ↔ ¬ 𝑋 ∈ (𝐾𝑠)))
27263adant3 1129 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝑋 ∈ (𝐼‘(𝐵𝑠)) ↔ ¬ 𝑋 ∈ (𝐾𝑠)))
2820, 27bitrd 278 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝑋 ∈ (𝐼𝑡) ↔ ¬ 𝑋 ∈ (𝐾𝑠)))
297, 17, 28rexxfrd2 5417 . 2 (𝜑 → (∃𝑡 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑡) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾𝑠)))
303, 29bitrid 282 1 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾𝑠)))
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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