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Theorem clsneifv4 40829
 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 3889 . 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 40824 . . . . . 6 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
9 elmapi 8413 . . . . . 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 6829 . . . 4 (𝜑 → (𝐾𝑆) ∈ 𝒫 𝐵)
1312elpwid 4508 . . 3 (𝜑 → (𝐾𝑆) ⊆ 𝐵)
14 sseqin2 4142 . . 3 ((𝐾𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐾𝑆)) = (𝐾𝑆))
1513, 14sylib 221 . 2 (𝜑 → (𝐵 ∩ (𝐾𝑆)) = (𝐾𝑆))
167adantr 484 . . . 4 ((𝜑𝑥𝐵) → 𝐾𝐻𝑁)
17 simpr 488 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
1811adantr 484 . . . 4 ((𝜑𝑥𝐵) → 𝑆 ∈ 𝒫 𝐵)
192, 3, 4, 5, 6, 16, 17, 18clsneiel1 40826 . . 3 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑥)))
2019rabbidva 3425 . 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 399   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497   class class class wbr 5030   ↦ cmpt 5110   ∘ ccom 5523  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   ↑m cmap 8391 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 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 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 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7673  df-2nd 7674  df-map 8393 This theorem is referenced by: (None)
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