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Theorem ntrclsneine0lem 41674
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 6774 . . . 4 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
21eleq2d 2824 . . 3 (𝑠 = 𝑡 → (𝑋 ∈ (𝐼𝑠) ↔ 𝑋 ∈ (𝐼𝑡)))
32cbvrexvw 3384 . 2 (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ ∃𝑡 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑡))
4 ntrcls.d . . . . 5 𝐷 = (𝑂𝐵)
5 ntrcls.r . . . . 5 (𝜑𝐼𝐷𝐾)
64, 5ntrclsrcomplex 41645 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
76adantr 481 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
84, 5ntrclsrcomplex 41645 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
98adantr 481 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
10 difeq2 4051 . . . . . 6 (𝑠 = (𝐵𝑡) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
1110adantl 482 . . . . 5 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
12 elpwi 4542 . . . . . . 7 (𝑡 ∈ 𝒫 𝐵𝑡𝐵)
13 dfss4 4192 . . . . . . 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 2779 . . . 4 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → 𝑡 = (𝐵𝑠))
179, 16rspcedeq2vd 3567 . . 3 ((𝜑𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠))
18 fveq2 6774 . . . . . 6 (𝑡 = (𝐵𝑠) → (𝐼𝑡) = (𝐼‘(𝐵𝑠)))
1918eleq2d 2824 . . . . 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 481 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐼𝐷𝐾)
23 ntrclslem0.x . . . . . . 7 (𝜑𝑋𝐵)
2423adantr 481 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
25 simpr 485 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
2621, 4, 22, 24, 25ntrclselnel2 41668 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼‘(𝐵𝑠)) ↔ ¬ 𝑋 ∈ (𝐾𝑠)))
27263adant3 1131 . . . 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 5336 . 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 396  w3a 1086   = wceq 1539  wcel 2106  wrex 3065  Vcvv 3432  cdif 3884  wss 3887  𝒫 cpw 4533   class class class wbr 5074  cmpt 5157  cfv 6433  (class class class)co 7275  m cmap 8615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617
This theorem is referenced by:  ntrclsneine0  41675
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