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Theorem clsneifv4 42325
Description: Value of the closure (interior) function in terms of the neighborhoods (convergents) function. (Contributed by RP, 27-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)
clsneifv.s (𝜑𝑆 ∈ 𝒫 𝐵)
Assertion
Ref Expression
clsneifv4 (𝜑 → (𝐾𝑆) = {𝑥𝐵 ∣ ¬ (𝐵𝑆) ∈ (𝑁𝑥)})
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑥   𝐵,𝑛,𝑜,𝑝,𝑥   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,𝑥   𝑛,𝐾,𝑜,𝑝   𝑖,𝑁,𝑗,𝑘,𝑙   𝑛,𝑁,𝑜,𝑝   𝑆,𝑚,𝑥   𝑆,𝑜   𝜑,𝑖,𝑗,𝑘,𝑙,𝑥   𝜑,𝑛,𝑜,𝑝
Allowed substitution hints:   𝜑(𝑚)   𝐷(𝑥)   𝑃(𝑥,𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑆(𝑖,𝑗,𝑘,𝑛,𝑝,𝑙)   𝐹(𝑥,𝑚)   𝐻(𝑥,𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑁(𝑥,𝑚)   𝑂(𝑥,𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)

Proof of Theorem clsneifv4
StepHypRef Expression
1 dfin5 3916 . 2 (𝐵 ∩ (𝐾𝑆)) = {𝑥𝐵𝑥 ∈ (𝐾𝑆)}
2 clsnei.o . . . . . . 7 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
3 clsnei.p . . . . . . 7 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
4 clsnei.d . . . . . . 7 𝐷 = (𝑃𝐵)
5 clsnei.f . . . . . . 7 𝐹 = (𝒫 𝐵𝑂𝐵)
6 clsnei.h . . . . . . 7 𝐻 = (𝐹𝐷)
7 clsnei.r . . . . . . 7 (𝜑𝐾𝐻𝑁)
82, 3, 4, 5, 6, 7clsneikex 42320 . . . . . 6 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
9 elmapi 8783 . . . . . 6 (𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
108, 9syl 17 . . . . 5 (𝜑𝐾:𝒫 𝐵⟶𝒫 𝐵)
11 clsneifv.s . . . . 5 (𝜑𝑆 ∈ 𝒫 𝐵)
1210, 11ffvelcdmd 7032 . . . 4 (𝜑 → (𝐾𝑆) ∈ 𝒫 𝐵)
1312elpwid 4567 . . 3 (𝜑 → (𝐾𝑆) ⊆ 𝐵)
14 sseqin2 4173 . . 3 ((𝐾𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐾𝑆)) = (𝐾𝑆))
1513, 14sylib 217 . 2 (𝜑 → (𝐵 ∩ (𝐾𝑆)) = (𝐾𝑆))
167adantr 481 . . . 4 ((𝜑𝑥𝐵) → 𝐾𝐻𝑁)
17 simpr 485 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
1811adantr 481 . . . 4 ((𝜑𝑥𝐵) → 𝑆 ∈ 𝒫 𝐵)
192, 3, 4, 5, 6, 16, 17, 18clsneiel1 42322 . . 3 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑥)))
2019rabbidva 3412 . 2 (𝜑 → {𝑥𝐵𝑥 ∈ (𝐾𝑆)} = {𝑥𝐵 ∣ ¬ (𝐵𝑆) ∈ (𝑁𝑥)})
211, 15, 203eqtr3a 2800 1 (𝜑 → (𝐾𝑆) = {𝑥𝐵 ∣ ¬ (𝐵𝑆) ∈ (𝑁𝑥)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  {crab 3405  Vcvv 3443  cdif 3905  cin 3907  wss 3908  𝒫 cpw 4558   class class class wbr 5103  cmpt 5186  ccom 5635  wf 6489  cfv 6493  (class class class)co 7353  cmpo 7355  m cmap 8761
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7917  df-2nd 7918  df-map 8763
This theorem is referenced by: (None)
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