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Theorem clsneiel2 42473
Description: If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then membership in the closure of the complement of a subset is equivalent to the subset not being a neighborhood of the point. (Contributed by RP, 7-Jun-2021.)
Hypotheses
Ref Expression
clsnei.o 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
clsnei.p 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
clsnei.d 𝐷 = (𝑃𝐵)
clsnei.f 𝐹 = (𝒫 𝐵𝑂𝐵)
clsnei.h 𝐻 = (𝐹𝐷)
clsnei.r (𝜑𝐾𝐻𝑁)
clsneiel.x (𝜑𝑋𝐵)
clsneiel.s (𝜑𝑆 ∈ 𝒫 𝐵)
Assertion
Ref Expression
clsneiel2 (𝜑 → (𝑋 ∈ (𝐾‘(𝐵𝑆)) ↔ ¬ 𝑆 ∈ (𝑁𝑋)))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚   𝐵,𝑛,𝑜,𝑝   𝐷,𝑖,𝑗,𝑘,𝑙,𝑚   𝐷,𝑛,𝑜,𝑝   𝑖,𝐹,𝑗,𝑘,𝑙   𝑛,𝐹,𝑜,𝑝   𝑖,𝐾,𝑗,𝑘,𝑙,𝑚   𝑛,𝐾,𝑜,𝑝   𝑖,𝑁,𝑗,𝑘,𝑙   𝑛,𝑁,𝑜,𝑝   𝑆,𝑚   𝑆,𝑜   𝑋,𝑙,𝑚   𝜑,𝑖,𝑗,𝑘,𝑙   𝜑,𝑛,𝑜,𝑝
Allowed substitution hints:   𝜑(𝑚)   𝑃(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑆(𝑖,𝑗,𝑘,𝑛,𝑝,𝑙)   𝐹(𝑚)   𝐻(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑁(𝑚)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑋(𝑖,𝑗,𝑘,𝑛,𝑜,𝑝)

Proof of Theorem clsneiel2
StepHypRef Expression
1 clsnei.o . . 3 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
2 clsnei.p . . 3 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
3 clsnei.d . . 3 𝐷 = (𝑃𝐵)
4 clsnei.f . . 3 𝐹 = (𝒫 𝐵𝑂𝐵)
5 clsnei.h . . 3 𝐻 = (𝐹𝐷)
6 clsnei.r . . 3 (𝜑𝐾𝐻𝑁)
7 clsneiel.x . . 3 (𝜑𝑋𝐵)
83, 5, 6clsneircomplex 42467 . . 3 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
91, 2, 3, 4, 5, 6, 7, 8clsneiel1 42472 . 2 (𝜑 → (𝑋 ∈ (𝐾‘(𝐵𝑆)) ↔ ¬ (𝐵 ∖ (𝐵𝑆)) ∈ (𝑁𝑋)))
10 clsneiel.s . . . . . 6 (𝜑𝑆 ∈ 𝒫 𝐵)
1110elpwid 4573 . . . . 5 (𝜑𝑆𝐵)
12 dfss4 4222 . . . . 5 (𝑆𝐵 ↔ (𝐵 ∖ (𝐵𝑆)) = 𝑆)
1311, 12sylib 217 . . . 4 (𝜑 → (𝐵 ∖ (𝐵𝑆)) = 𝑆)
1413eleq1d 2819 . . 3 (𝜑 → ((𝐵 ∖ (𝐵𝑆)) ∈ (𝑁𝑋) ↔ 𝑆 ∈ (𝑁𝑋)))
1514notbid 318 . 2 (𝜑 → (¬ (𝐵 ∖ (𝐵𝑆)) ∈ (𝑁𝑋) ↔ ¬ 𝑆 ∈ (𝑁𝑋)))
169, 15bitrd 279 1 (𝜑 → (𝑋 ∈ (𝐾‘(𝐵𝑆)) ↔ ¬ 𝑆 ∈ (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1542  wcel 2107  {crab 3406  Vcvv 3447  cdif 3911  wss 3914  𝒫 cpw 4564   class class class wbr 5109  cmpt 5192  ccom 5641  cfv 6500  (class class class)co 7361  cmpo 7363  m cmap 8771
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-map 8773
This theorem is referenced by:  clsneifv3  42474
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