Theorem neicvgnvo 44128

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 5885	. . . 4 ⊢ ◡𝐻 = ◡(𝐹 ∘ (𝐷 ∘ 𝐺))
3		cnvco 5896	. . . 4 ⊢ ◡(𝐹 ∘ (𝐷 ∘ 𝐺)) = (◡(𝐷 ∘ 𝐺) ∘ ◡𝐹)
4		cnvco 5896	. . . . 5 ⊢ ◡(𝐷 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐷)
5	4	coeq1i 5870	. . . 4 ⊢ (◡(𝐷 ∘ 𝐺) ∘ ◡𝐹) = ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹)
6	2, 3, 5	3eqtri 2769	. . 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 44125	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
11	10	pwexd 5379	. . . . . 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 44027	. . . . 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 44033	. . . . 5 ⊢ (𝜑 → ◡𝐷 = 𝐷)
17	14, 16	coeq12d 5875	. . . 4 ⊢ (𝜑 → (◡𝐺 ∘ ◡𝐷) = (𝐹 ∘ 𝐷))
18	7, 11, 10, 13, 12	fsovcnvd 44027	. . . 4 ⊢ (𝜑 → ◡𝐹 = 𝐺)
19	17, 18	coeq12d 5875	. . 3 ⊢ (𝜑 → ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹) = ((𝐹 ∘ 𝐷) ∘ 𝐺))
20	6, 19	eqtrid 2789	. 2 ⊢ (𝜑 → ◡𝐻 = ((𝐹 ∘ 𝐷) ∘ 𝐺))
21		coass 6285	. . 3 ⊢ ((𝐹 ∘ 𝐷) ∘ 𝐺) = (𝐹 ∘ (𝐷 ∘ 𝐺))
22	21, 1	eqtr4i 2768	. 2 ⊢ ((𝐹 ∘ 𝐷) ∘ 𝐺) = 𝐻
23	20, 22	eqtrdi 2793	1 ⊢ (𝜑 → ◡𝐻 = 𝐻)

Syntax hints:   → wi 4   = 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:  neicvgnvor  44129