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Theorem ntrclsneine0lem 41212
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 6675 . . . 4 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
21eleq2d 2818 . . 3 (𝑠 = 𝑡 → (𝑋 ∈ (𝐼𝑠) ↔ 𝑋 ∈ (𝐼𝑡)))
32cbvrexvw 3350 . 2 (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ ∃𝑡 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑡))
4 ntrcls.d . . . . 5 𝐷 = (𝑂𝐵)
5 ntrcls.r . . . . 5 (𝜑𝐼𝐷𝐾)
64, 5ntrclsrcomplex 41183 . . . 4 (𝜑 → (𝐵𝑠) ∈ 𝒫 𝐵)
76adantr 484 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐵𝑠) ∈ 𝒫 𝐵)
84, 5ntrclsrcomplex 41183 . . . . 5 (𝜑 → (𝐵𝑡) ∈ 𝒫 𝐵)
98adantr 484 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐵𝑡) ∈ 𝒫 𝐵)
10 difeq2 4008 . . . . . 6 (𝑠 = (𝐵𝑡) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
1110adantl 485 . . . . 5 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝐵𝑠) = (𝐵 ∖ (𝐵𝑡)))
12 elpwi 4498 . . . . . . 7 (𝑡 ∈ 𝒫 𝐵𝑡𝐵)
13 dfss4 4150 . . . . . . 7 (𝑡𝐵 ↔ (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1412, 13sylib 221 . . . . . 6 (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1514ad2antlr 727 . . . . 5 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → (𝐵 ∖ (𝐵𝑡)) = 𝑡)
1611, 15eqtr2d 2774 . . . 4 (((𝜑𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵𝑡)) → 𝑡 = (𝐵𝑠))
179, 16rspcedeq2vd 3534 . . 3 ((𝜑𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠))
18 fveq2 6675 . . . . . 6 (𝑡 = (𝐵𝑠) → (𝐼𝑡) = (𝐼‘(𝐵𝑠)))
1918eleq2d 2818 . . . . 5 (𝑡 = (𝐵𝑠) → (𝑋 ∈ (𝐼𝑡) ↔ 𝑋 ∈ (𝐼‘(𝐵𝑠))))
20193ad2ant3 1136 . . . 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 484 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐼𝐷𝐾)
23 ntrclslem0.x . . . . . . 7 (𝜑𝑋𝐵)
2423adantr 484 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
25 simpr 488 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
2621, 4, 22, 24, 25ntrclselnel2 41206 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼‘(𝐵𝑠)) ↔ ¬ 𝑋 ∈ (𝐾𝑠)))
27263adant3 1133 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝑋 ∈ (𝐼‘(𝐵𝑠)) ↔ ¬ 𝑋 ∈ (𝐾𝑠)))
2820, 27bitrd 282 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵𝑠)) → (𝑋 ∈ (𝐼𝑡) ↔ ¬ 𝑋 ∈ (𝐾𝑠)))
297, 17, 28rexxfrd2 5281 . 2 (𝜑 → (∃𝑡 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑡) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾𝑠)))
303, 29syl5bb 286 1 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾𝑠)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2113  wrex 3054  Vcvv 3398  cdif 3841  wss 3844  𝒫 cpw 4489   class class class wbr 5031  cmpt 5111  cfv 6340  (class class class)co 7171  m cmap 8438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7480
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3683  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7174  df-oprab 7175  df-mpo 7176  df-1st 7715  df-2nd 7716  df-map 8440
This theorem is referenced by:  ntrclsneine0  41213
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