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Theorem clsneifv3 40648
 Description: Value of the neighborhoods (convergents) in terms of the closure (interior) 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.x (𝜑𝑋𝐵)
Assertion
Ref Expression
clsneifv3 (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵𝑠))})
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠   𝐵,𝑛,𝑜,𝑝,𝑠   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,𝑠   𝑛,𝑁,𝑜,𝑝   𝑋,𝑙,𝑚,𝑠   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠   𝜑,𝑛,𝑜,𝑝
Allowed substitution hints:   𝜑(𝑚)   𝐷(𝑠)   𝑃(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑠,𝑝,𝑙)   𝐹(𝑚,𝑠)   𝐻(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑠,𝑝,𝑙)   𝐾(𝑠)   𝑁(𝑚)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑠,𝑝,𝑙)   𝑋(𝑖,𝑗,𝑘,𝑛,𝑜,𝑝)

Proof of Theorem clsneifv3
StepHypRef Expression
1 dfin5 3926 . 2 (𝒫 𝐵 ∩ (𝑁𝑋)) = {𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)}
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, 7clsneinex 40645 . . . . . 6 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
9 elmapi 8411 . . . . . 6 (𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
108, 9syl 17 . . . . 5 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
11 clsneifv.x . . . . 5 (𝜑𝑋𝐵)
1210, 11ffvelrnd 6833 . . . 4 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
1312elpwid 4531 . . 3 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
14 sseqin2 4175 . . 3 ((𝑁𝑋) ⊆ 𝒫 𝐵 ↔ (𝒫 𝐵 ∩ (𝑁𝑋)) = (𝑁𝑋))
1513, 14sylib 221 . 2 (𝜑 → (𝒫 𝐵 ∩ (𝑁𝑋)) = (𝑁𝑋))
167adantr 484 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐾𝐻𝑁)
1711adantr 484 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
18 simpr 488 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
192, 3, 4, 5, 6, 16, 17, 18clsneiel2 40647 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐾‘(𝐵𝑠)) ↔ ¬ 𝑠 ∈ (𝑁𝑋)))
2019con2bid 358 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑠 ∈ (𝑁𝑋) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵𝑠))))
2120rabbidva 3463 . 2 (𝜑 → {𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵𝑠))})
221, 15, 213eqtr3a 2883 1 (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵𝑠))})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  {crab 3136  Vcvv 3479   ∖ cdif 3915   ∩ cin 3917   ⊆ wss 3918  𝒫 cpw 4520   class class class wbr 5047   ↦ cmpt 5127   ∘ ccom 5540  ⟶wf 6332  ‘cfv 6336  (class class class)co 7138   ∈ cmpo 7140   ↑m cmap 8389 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-1st 7672  df-2nd 7673  df-map 8391 This theorem is referenced by: (None)
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