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Theorem clsneifv4 39191
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 3777 . 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 39186 . . . . . 6 (𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
9 elmapi 8117 . . . . . 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, 11ffvelrnd 6586 . . . 4 (𝜑 → (𝐾𝑆) ∈ 𝒫 𝐵)
1312elpwid 4361 . . 3 (𝜑 → (𝐾𝑆) ⊆ 𝐵)
14 sseqin2 4015 . . 3 ((𝐾𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐾𝑆)) = (𝐾𝑆))
1513, 14sylib 210 . 2 (𝜑 → (𝐵 ∩ (𝐾𝑆)) = (𝐾𝑆))
167adantr 473 . . . 4 ((𝜑𝑥𝐵) → 𝐾𝐻𝑁)
17 simpr 478 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
1811adantr 473 . . . 4 ((𝜑𝑥𝐵) → 𝑆 ∈ 𝒫 𝐵)
192, 3, 4, 5, 6, 16, 17, 18clsneiel1 39188 . . 3 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑥)))
2019rabbidva 3372 . 2 (𝜑 → {𝑥𝐵𝑥 ∈ (𝐾𝑆)} = {𝑥𝐵 ∣ ¬ (𝐵𝑆) ∈ (𝑁𝑥)})
211, 15, 203eqtr3a 2857 1 (𝜑 → (𝐾𝑆) = {𝑥𝐵 ∣ ¬ (𝐵𝑆) ∈ (𝑁𝑥)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385   = wceq 1653  wcel 2157  {crab 3093  Vcvv 3385  cdif 3766  cin 3768  wss 3769  𝒫 cpw 4349   class class class wbr 4843  cmpt 4922  ccom 5316  wf 6097  cfv 6101  (class class class)co 6878  cmpt2 6880  𝑚 cmap 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-map 8097
This theorem is referenced by: (None)
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