Theorem clsneicnv 44101

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 5841	. . 3 ⊢ ◡𝐻 = ◡(𝐹 ∘ 𝐷)
3		cnvco 5852	. . 3 ⊢ ◡(𝐹 ∘ 𝐷) = (◡𝐷 ∘ ◡𝐹)
4	2, 3	eqtri 2753	. 2 ⊢ ◡𝐻 = (◡𝐷 ∘ ◡𝐹)
5		clsnei.d	. . . 4 ⊢ 𝐷 = (𝑃‘𝐵)
6		clsnei.r	. . . 4 ⊢ (𝜑 → 𝐾𝐻𝑁)
7	5, 1, 6	clsneibex 44098	. . 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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
10	8, 5, 9	dssmapnvod 44016	. . . 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 5336	. . . . . 6 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
13	12	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
14		clsnei.f	. . . . 5 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
15		eqid 2730	. . . . 5 ⊢ (𝐵𝑂𝒫 𝐵) = (𝐵𝑂𝒫 𝐵)
16	11, 13, 9, 14, 15	fsovcnvd 44010	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → ◡𝐹 = (𝐵𝑂𝒫 𝐵))
17	10, 16	coeq12d 5831	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (◡𝐷 ∘ ◡𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
18	7, 17	mpdan 687	. 2 ⊢ (𝜑 → (◡𝐷 ∘ ◡𝐹) = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
19	4, 18	eqtrid 2777	1 ⊢ (𝜑 → ◡𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3408  Vcvv 3450   ∖ cdif 3914  𝒫 cpw 4566   class class class wbr 5110   ↦ cmpt 5191  ^◡ccnv 5640   ∘ ccom 5645  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392   ↑_m cmap 8802

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804

This theorem is referenced by: (None)