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Theorem clsneinex 41211
 Description: If closure and neighborhoods functions are related, the neighborhoods 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
clsneinex (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 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 clsneinex
StepHypRef Expression
1 clsnei.o . 2 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
2 clsnei.f . 2 𝐹 = (𝒫 𝐵𝑂𝐵)
3 clsnei.d . . . . 5 𝐷 = (𝑃𝐵)
4 clsnei.h . . . . 5 𝐻 = (𝐹𝐷)
5 clsnei.r . . . . 5 (𝜑𝐾𝐻𝑁)
63, 4, 5clsneibex 41206 . . . 4 (𝜑𝐵 ∈ V)
7 pwexg 5250 . . . . . . . 8 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
87adantl 485 . . . . . . 7 ((𝜑𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
9 simpr 488 . . . . . . 7 ((𝜑𝐵 ∈ V) → 𝐵 ∈ V)
101, 8, 9, 2fsovf1od 41118 . . . . . 6 ((𝜑𝐵 ∈ V) → 𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
11 f1ofn 6607 . . . . . 6 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → 𝐹 Fn (𝒫 𝐵m 𝒫 𝐵))
1210, 11syl 17 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐹 Fn (𝒫 𝐵m 𝒫 𝐵))
13 clsnei.p . . . . . . 7 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
1413, 3, 9dssmapf1od 41123 . . . . . 6 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
15 f1of 6606 . . . . . 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))
175adantr 484 . . . . . 6 ((𝜑𝐵 ∈ V) → 𝐾𝐻𝑁)
184breqi 5041 . . . . . 6 (𝐾𝐻𝑁𝐾(𝐹𝐷)𝑁)
1917, 18sylib 221 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐾(𝐹𝐷)𝑁)
2012, 16, 19brcoffn 41134 . . . 4 ((𝜑𝐵 ∈ V) → (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁))
216, 20mpdan 686 . . 3 (𝜑 → (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁))
2221simprd 499 . 2 (𝜑 → (𝐷𝐾)𝐹𝑁)
231, 2, 22ntrneinex 41181 1 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3074  Vcvv 3409   ∖ cdif 3857  𝒫 cpw 4497   class class class wbr 5035   ↦ cmpt 5115   ∘ ccom 5531   Fn wfn 6334  ⟶wf 6335  –1-1-onto→wf1o 6338  ‘cfv 6339  (class class class)co 7155   ∈ cmpo 7157   ↑m cmap 8421 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 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 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 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7698  df-2nd 7699  df-map 8423 This theorem is referenced by:  clsneifv3  41214
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