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Theorem clsneicnv 44118
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 5885 . . 3 𝐻 = (𝐹𝐷)
3 cnvco 5896 . . 3 (𝐹𝐷) = (𝐷𝐹)
42, 3eqtri 2765 . 2 𝐻 = (𝐷𝐹)
5 clsnei.d . . . 4 𝐷 = (𝑃𝐵)
6 clsnei.r . . . 4 (𝜑𝐾𝐻𝑁)
75, 1, 6clsneibex 44115 . . 3 (𝜑𝐵 ∈ V)
8 clsnei.p . . . . 5 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
9 simpr 484 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐵 ∈ V)
108, 5, 9dssmapnvod 44033 . . . 4 ((𝜑𝐵 ∈ V) → 𝐷 = 𝐷)
11 clsnei.o . . . . 5 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
12 pwexg 5378 . . . . . 6 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
1312adantl 481 . . . . 5 ((𝜑𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
14 clsnei.f . . . . 5 𝐹 = (𝒫 𝐵𝑂𝐵)
15 eqid 2737 . . . . 5 (𝐵𝑂𝒫 𝐵) = (𝐵𝑂𝒫 𝐵)
1611, 13, 9, 14, 15fsovcnvd 44027 . . . 4 ((𝜑𝐵 ∈ V) → 𝐹 = (𝐵𝑂𝒫 𝐵))
1710, 16coeq12d 5875 . . 3 ((𝜑𝐵 ∈ V) → (𝐷𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
187, 17mpdan 687 . 2 (𝜑 → (𝐷𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
194, 18eqtrid 2789 1 (𝜑𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  cdif 3948  𝒫 cpw 4600   class class class wbr 5143  cmpt 5225  ccnv 5684  ccom 5689  cfv 6561  (class class class)co 7431  cmpo 7433  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by: (None)
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