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Theorem clsneifv4 40339
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 3941 . 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 40334 . . . . . 6 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
9 elmapi 8417 . . . . . 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 6844 . . . 4 (𝜑 → (𝐾𝑆) ∈ 𝒫 𝐵)
1312elpwid 4549 . . 3 (𝜑 → (𝐾𝑆) ⊆ 𝐵)
14 sseqin2 4189 . . 3 ((𝐾𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐾𝑆)) = (𝐾𝑆))
1513, 14sylib 219 . 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 40336 . . 3 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑥)))
2019rabbidva 3476 . 2 (𝜑 → {𝑥𝐵𝑥 ∈ (𝐾𝑆)} = {𝑥𝐵 ∣ ¬ (𝐵𝑆) ∈ (𝑁𝑥)})
211, 15, 203eqtr3a 2877 1 (𝜑 → (𝐾𝑆) = {𝑥𝐵 ∣ ¬ (𝐵𝑆) ∈ (𝑁𝑥)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  cdif 3930  cin 3932  wss 3933  𝒫 cpw 4535   class class class wbr 5057  cmpt 5137  ccom 5552  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  m cmap 8395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397
This theorem is referenced by: (None)
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