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Theorem clsneikex 41182
 Description: If closure and neighborhoods functions are related, the closure function exists. (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)
Assertion
Ref Expression
clsneikex (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚   𝐵,𝑛,𝑜,𝑝   𝜑,𝑖,𝑗,𝑘,𝑙   𝜑,𝑛,𝑜,𝑝
Allowed substitution hints:   𝜑(𝑚)   𝐷(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑃(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝐹(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝐻(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝐾(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)

Proof of Theorem clsneikex
StepHypRef Expression
1 clsnei.p . 2 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
2 clsnei.d . 2 𝐷 = (𝑃𝐵)
3 clsnei.h . . . . 5 𝐻 = (𝐹𝐷)
4 clsnei.r . . . . 5 (𝜑𝐾𝐻𝑁)
52, 3, 4clsneibex 41178 . . . 4 (𝜑𝐵 ∈ V)
6 clsnei.o . . . . . . 7 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
7 pwexg 5247 . . . . . . . 8 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
87adantl 485 . . . . . . 7 ((𝜑𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
9 simpr 488 . . . . . . 7 ((𝜑𝐵 ∈ V) → 𝐵 ∈ V)
10 clsnei.f . . . . . . 7 𝐹 = (𝒫 𝐵𝑂𝐵)
116, 8, 9, 10fsovf1od 41090 . . . . . 6 ((𝜑𝐵 ∈ V) → 𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
12 f1ofn 6603 . . . . . 6 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → 𝐹 Fn (𝒫 𝐵m 𝒫 𝐵))
1311, 12syl 17 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐹 Fn (𝒫 𝐵m 𝒫 𝐵))
141, 2, 9dssmapf1od 41095 . . . . . 6 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
15 f1of 6602 . . . . . 6 (𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵) → 𝐷:(𝒫 𝐵m 𝒫 𝐵)⟶(𝒫 𝐵m 𝒫 𝐵))
1614, 15syl 17 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵m 𝒫 𝐵)⟶(𝒫 𝐵m 𝒫 𝐵))
174adantr 484 . . . . . 6 ((𝜑𝐵 ∈ V) → 𝐾𝐻𝑁)
183breqi 5038 . . . . . 6 (𝐾𝐻𝑁𝐾(𝐹𝐷)𝑁)
1917, 18sylib 221 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐾(𝐹𝐷)𝑁)
2013, 16, 19brcoffn 41106 . . . 4 ((𝜑𝐵 ∈ V) → (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁))
215, 20mpdan 686 . . 3 (𝜑 → (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁))
2221simpld 498 . 2 (𝜑𝐾𝐷(𝐷𝐾))
231, 2, 22ntrclsiex 41129 1 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3074  Vcvv 3409   ∖ cdif 3855  𝒫 cpw 4494   class class class wbr 5032   ↦ cmpt 5112   ∘ ccom 5528   Fn wfn 6330  ⟶wf 6331  –1-1-onto→wf1o 6334  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152   ↑m cmap 8416 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 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-map 8418 This theorem is referenced by:  clsneifv4  41187
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