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Theorem clsneicnv 37882
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 5257 . . 3 𝐻 = (𝐹𝐷)
3 cnvco 5268 . . 3 (𝐹𝐷) = (𝐷𝐹)
42, 3eqtri 2643 . 2 𝐻 = (𝐷𝐹)
5 clsnei.d . . . 4 𝐷 = (𝑃𝐵)
6 clsnei.r . . . 4 (𝜑𝐾𝐻𝑁)
75, 1, 6clsneibex 37879 . . 3 (𝜑𝐵 ∈ V)
8 clsnei.p . . . . 5 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
9 simpr 477 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐵 ∈ V)
108, 5, 9dssmapnvod 37793 . . . 4 ((𝜑𝐵 ∈ V) → 𝐷 = 𝐷)
11 clsnei.o . . . . 5 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
12 pwexg 4810 . . . . . 6 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
1312adantl 482 . . . . 5 ((𝜑𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
14 clsnei.f . . . . 5 𝐹 = (𝒫 𝐵𝑂𝐵)
15 eqid 2621 . . . . 5 (𝐵𝑂𝒫 𝐵) = (𝐵𝑂𝒫 𝐵)
1611, 13, 9, 14, 15fsovcnvd 37787 . . . 4 ((𝜑𝐵 ∈ V) → 𝐹 = (𝐵𝑂𝒫 𝐵))
1710, 16coeq12d 5246 . . 3 ((𝜑𝐵 ∈ V) → (𝐷𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
187, 17mpdan 701 . 2 (𝜑 → (𝐷𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
194, 18syl5eq 2667 1 (𝜑𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  cdif 3552  𝒫 cpw 4130   class class class wbr 4613  cmpt 4673  ccnv 5073  ccom 5078  cfv 5847  (class class class)co 6604  cmpt2 6606  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804
This theorem is referenced by: (None)
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