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Theorem clsneicnv 38874
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 5440 . . 3 𝐻 = (𝐹𝐷)
3 cnvco 5451 . . 3 (𝐹𝐷) = (𝐷𝐹)
42, 3eqtri 2770 . 2 𝐻 = (𝐷𝐹)
5 clsnei.d . . . 4 𝐷 = (𝑃𝐵)
6 clsnei.r . . . 4 (𝜑𝐾𝐻𝑁)
75, 1, 6clsneibex 38871 . . 3 (𝜑𝐵 ∈ V)
8 clsnei.p . . . . 5 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
9 simpr 479 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐵 ∈ V)
108, 5, 9dssmapnvod 38785 . . . 4 ((𝜑𝐵 ∈ V) → 𝐷 = 𝐷)
11 clsnei.o . . . . 5 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
12 pwexg 4987 . . . . . 6 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
1312adantl 473 . . . . 5 ((𝜑𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
14 clsnei.f . . . . 5 𝐹 = (𝒫 𝐵𝑂𝐵)
15 eqid 2748 . . . . 5 (𝐵𝑂𝒫 𝐵) = (𝐵𝑂𝒫 𝐵)
1611, 13, 9, 14, 15fsovcnvd 38779 . . . 4 ((𝜑𝐵 ∈ V) → 𝐹 = (𝐵𝑂𝒫 𝐵))
1710, 16coeq12d 5430 . . 3 ((𝜑𝐵 ∈ V) → (𝐷𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
187, 17mpdan 705 . 2 (𝜑 → (𝐷𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
194, 18syl5eq 2794 1 (𝜑𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wcel 2127  {crab 3042  Vcvv 3328  cdif 3700  𝒫 cpw 4290   class class class wbr 4792  cmpt 4869  ccnv 5253  ccom 5258  cfv 6037  (class class class)co 6801  cmpt2 6803  𝑚 cmap 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-1st 7321  df-2nd 7322  df-map 8013
This theorem is referenced by: (None)
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