Theorem clsneicnv 44859

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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
clsnei.p	⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))
clsnei.d	⊢ 𝐷 = (𝑃‘𝐵)
clsnei.f	⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
clsnei.h	⊢ 𝐻 = (𝐹 ∘ 𝐷)
clsnei.r	⊢ (𝜑 → 𝐾𝐻𝑁)

Assertion

Ref	Expression
clsneicnv	⊢ (𝜑 → ◡𝐻 = (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

Step	Hyp	Ref	Expression
1		clsnei.h	. . . 4 ⊢ 𝐻 = (𝐹 ∘ 𝐷)
2	1	cnveqi 5859	. . 3 ⊢ ◡𝐻 = ◡(𝐹 ∘ 𝐷)
3		cnvco 5874	. . 3 ⊢ ◡(𝐹 ∘ 𝐷) = (◡𝐷 ∘ ◡𝐹)
4	2, 3	eqtri 2785	. 2 ⊢ ◡𝐻 = (◡𝐷 ∘ ◡𝐹)
5		clsnei.d	. . . 4 ⊢ 𝐷 = (𝑃‘𝐵)
6		clsnei.r	. . . 4 ⊢ (𝜑 → 𝐾𝐻𝑁)
7	5, 1, 6	clsneibex 44856	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
8		clsnei.p	. . . . 5 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))
9		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
10	8, 5, 9	dssmapnvod 44774	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → ◡𝐷 = 𝐷)
11		clsnei.o	. . . . 5 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
12		pwexg 5348	. . . . . 6 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
13	12	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
14		clsnei.f	. . . . 5 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
15		eqid 2762	. . . . 5 ⊢ (𝐵𝑂𝒫 𝐵) = (𝐵𝑂𝒫 𝐵)
16	11, 13, 9, 14, 15	fsovcnvd 44768	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → ◡𝐹 = (𝐵𝑂𝒫 𝐵))
17	10, 16	coeq12d 5849	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (◡𝐷 ∘ ◡𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
18	7, 17	mpdan 699	. 2 ⊢ (𝜑 → (◡𝐷 ∘ ◡𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
19	4, 18	eqtrid 2809	1 ⊢ (𝜑 → ◡𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {crab 3415  Vcvv 3454   ∖ cdif 3901  𝒫 cpw 4561   class class class wbr 5108   ↦ cmpt 5191  ^◡ccnv 5659   ∘ ccom 5664  ‘cfv 6536  (class class class)co 7412   ∈ cmpo 7414   ↑_m cmap 8822

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-map 8824

This theorem is used by: (None)