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Theorem clsneiel2 40855
 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 40849 . . 3 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
91, 2, 3, 4, 5, 6, 7, 8clsneiel1 40854 . 2 (𝜑 → (𝑋 ∈ (𝐾‘(𝐵𝑆)) ↔ ¬ (𝐵 ∖ (𝐵𝑆)) ∈ (𝑁𝑋)))
10 clsneiel.s . . . . . 6 (𝜑𝑆 ∈ 𝒫 𝐵)
1110elpwid 4508 . . . . 5 (𝜑𝑆𝐵)
12 dfss4 4185 . . . . 5 (𝑆𝐵 ↔ (𝐵 ∖ (𝐵𝑆)) = 𝑆)
1311, 12sylib 221 . . . 4 (𝜑 → (𝐵 ∖ (𝐵𝑆)) = 𝑆)
1413eleq1d 2874 . . 3 (𝜑 → ((𝐵 ∖ (𝐵𝑆)) ∈ (𝑁𝑋) ↔ 𝑆 ∈ (𝑁𝑋)))
1514notbid 321 . 2 (𝜑 → (¬ (𝐵 ∖ (𝐵𝑆)) ∈ (𝑁𝑋) ↔ ¬ 𝑆 ∈ (𝑁𝑋)))
169, 15bitrd 282 1 (𝜑 → (𝑋 ∈ (𝐾‘(𝐵𝑆)) ↔ ¬ 𝑆 ∈ (𝑁𝑋)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  𝒫 cpw 4497   class class class wbr 5031   ↦ cmpt 5111   ∘ ccom 5524  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138   ↑m cmap 8392 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675  df-map 8394 This theorem is referenced by:  clsneifv3  40856
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