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Theorem clsneicnv 40745
Description: If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then the converse of the operator is known. (Contributed by RP, 5-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
clsneicnv (𝜑𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 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 clsneicnv
StepHypRef Expression
1 clsnei.h . . . 4 𝐻 = (𝐹𝐷)
21cnveqi 5722 . . 3 𝐻 = (𝐹𝐷)
3 cnvco 5733 . . 3 (𝐹𝐷) = (𝐷𝐹)
42, 3eqtri 2845 . 2 𝐻 = (𝐷𝐹)
5 clsnei.d . . . 4 𝐷 = (𝑃𝐵)
6 clsnei.r . . . 4 (𝜑𝐾𝐻𝑁)
75, 1, 6clsneibex 40742 . . 3 (𝜑𝐵 ∈ V)
8 clsnei.p . . . . 5 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
9 simpr 488 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐵 ∈ V)
108, 5, 9dssmapnvod 40656 . . . 4 ((𝜑𝐵 ∈ V) → 𝐷 = 𝐷)
11 clsnei.o . . . . 5 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
12 pwexg 5256 . . . . . 6 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
1312adantl 485 . . . . 5 ((𝜑𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
14 clsnei.f . . . . 5 𝐹 = (𝒫 𝐵𝑂𝐵)
15 eqid 2822 . . . . 5 (𝐵𝑂𝒫 𝐵) = (𝐵𝑂𝒫 𝐵)
1611, 13, 9, 14, 15fsovcnvd 40650 . . . 4 ((𝜑𝐵 ∈ V) → 𝐹 = (𝐵𝑂𝒫 𝐵))
1710, 16coeq12d 5712 . . 3 ((𝜑𝐵 ∈ V) → (𝐷𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
187, 17mpdan 686 . 2 (𝜑 → (𝐷𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
194, 18syl5eq 2869 1 (𝜑𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  {crab 3134  Vcvv 3469  cdif 3905  𝒫 cpw 4511   class class class wbr 5042  cmpt 5122  ccnv 5531  ccom 5536  cfv 6334  (class class class)co 7140  cmpo 7142  m cmap 8393
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-map 8395
This theorem is referenced by: (None)
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