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Theorem ntrclsneine0lem 43118
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 6891 . . . 4 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
21eleq2d 2818 . . 3 (𝑠 = 𝑡 → (𝑋 ∈ (𝐼𝑠) ↔ 𝑋 ∈ (𝐼𝑡)))
32cbvrexvw 3234 . 2 (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ ∃𝑡 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑡))
4 ntrcls.d . . . . 5 𝐷 = (𝑂𝐵)
5 ntrcls.r . . . . 5 (𝜑𝐼𝐷𝐾)
64, 5ntrclsrcomplex 43089 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
76adantr 480 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
84, 5ntrclsrcomplex 43089 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
98adantr 480 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
10 difeq2 4116 . . . . . 6 (𝑠 = (𝐵𝑡) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
1110adantl 481 . . . . 5 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
12 elpwi 4609 . . . . . . 7 (𝑡 ∈ 𝒫 𝐵𝑡𝐵)
13 dfss4 4258 . . . . . . 7 (𝑡𝐵 ↔ (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1412, 13sylib 217 . . . . . 6 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1514ad2antlr 724 . . . . 5 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1611, 15eqtr2d 2772 . . . 4 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → 𝑡 = (𝐵𝑠))
179, 16rspcedeq2vd 3619 . . 3 ((𝜑𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠))
18 fveq2 6891 . . . . . 6 (𝑡 = (𝐵𝑠) → (𝐼𝑡) = (𝐼‘(𝐵𝑠)))
1918eleq2d 2818 . . . . 5 (𝑡 = (𝐵𝑠) → (𝑋 ∈ (𝐼𝑡) ↔ 𝑋 ∈ (𝐼‘(𝐵𝑠))))
20193ad2ant3 1134 . . . 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 43112 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼‘(𝐵𝑠)) ↔ ¬ 𝑋 ∈ (𝐾𝑠)))
27263adant3 1131 . . . 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 5411 . 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 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wrex 3069  Vcvv 3473  cdif 3945  wss 3948  𝒫 cpw 4602   class class class wbr 5148  cmpt 5231  cfv 6543  (class class class)co 7412  m cmap 8824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-map 8826
This theorem is referenced by:  ntrclsneine0  43119
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