Theorem neicvgnvo 39182

Description: If neighborhood and convergent functions are related by operator 𝐻, it is its own converse function. (Contributed by RP, 11-Jun-2021.)

Hypotheses

Ref	Expression
neicvg.o	⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
neicvg.p	⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))
neicvg.d	⊢ 𝐷 = (𝑃‘𝐵)
neicvg.f	⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
neicvg.g	⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)
neicvg.h	⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))
neicvg.r	⊢ (𝜑 → 𝑁𝐻𝑀)

Assertion

Ref	Expression
neicvgnvo	⊢ (𝜑 → ◡𝐻 = 𝐻)

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)   𝐺(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝐻(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑀(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 neicvgnvo

Step	Hyp	Ref	Expression
1		neicvg.h	. . . . 5 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))
2	1	cnveqi 5498	. . . 4 ⊢ ◡𝐻 = ◡(𝐹 ∘ (𝐷 ∘ 𝐺))
3		cnvco 5509	. . . 4 ⊢ ◡(𝐹 ∘ (𝐷 ∘ 𝐺)) = (◡(𝐷 ∘ 𝐺) ∘ ◡𝐹)
4		cnvco 5509	. . . . 5 ⊢ ◡(𝐷 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐷)
5	4	coeq1i 5483	. . . 4 ⊢ (◡(𝐷 ∘ 𝐺) ∘ ◡𝐹) = ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹)
6	2, 3, 5	3eqtri 2823	. . 3 ⊢ ◡𝐻 = ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹)
7		neicvg.o	. . . . . 6 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
8		neicvg.d	. . . . . . 7 ⊢ 𝐷 = (𝑃‘𝐵)
9		neicvg.r	. . . . . . 7 ⊢ (𝜑 → 𝑁𝐻𝑀)
10	8, 1, 9	neicvgbex 39179	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
11	10	pwexd 5047	. . . . . 6 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
12		neicvg.g	. . . . . 6 ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)
13		neicvg.f	. . . . . 6 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
14	7, 10, 11, 12, 13	fsovcnvd 39077	. . . . 5 ⊢ (𝜑 → ◡𝐺 = 𝐹)
15		neicvg.p	. . . . . 6 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))
16	15, 8, 10	dssmapnvod 39083	. . . . 5 ⊢ (𝜑 → ◡𝐷 = 𝐷)
17	14, 16	coeq12d 5488	. . . 4 ⊢ (𝜑 → (◡𝐺 ∘ ◡𝐷) = (𝐹 ∘ 𝐷))
18	7, 11, 10, 13, 12	fsovcnvd 39077	. . . 4 ⊢ (𝜑 → ◡𝐹 = 𝐺)
19	17, 18	coeq12d 5488	. . 3 ⊢ (𝜑 → ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹) = ((𝐹 ∘ 𝐷) ∘ 𝐺))
20	6, 19	syl5eq 2843	. 2 ⊢ (𝜑 → ◡𝐻 = ((𝐹 ∘ 𝐷) ∘ 𝐺))
21		coass 5871	. . 3 ⊢ ((𝐹 ∘ 𝐷) ∘ 𝐺) = (𝐹 ∘ (𝐷 ∘ 𝐺))
22	21, 1	eqtr4i 2822	. 2 ⊢ ((𝐹 ∘ 𝐷) ∘ 𝐺) = 𝐻
23	20, 22	syl6eq 2847	1 ⊢ (𝜑 → ◡𝐻 = 𝐻)

Syntax hints:   → wi 4   = wceq 1653   ∈ wcel 2157  {crab 3091  Vcvv 3383   ∖ cdif 3764  𝒫 cpw 4347   class class class wbr 4841   ↦ cmpt 4920  ^◡ccnv 5309   ∘ ccom 5314  ‘cfv 6099  (class class class)co 6876   ↦ cmpt2 6878   ↑_𝑚 cmap 8093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-map 8095

This theorem is referenced by:  neicvgnvor  39183