Theorem neicvgnvo 42948

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 5874	. . . 4 ⊢ ◡𝐻 = ◡(𝐹 ∘ (𝐷 ∘ 𝐺))
3		cnvco 5885	. . . 4 ⊢ ◡(𝐹 ∘ (𝐷 ∘ 𝐺)) = (◡(𝐷 ∘ 𝐺) ∘ ◡𝐹)
4		cnvco 5885	. . . . 5 ⊢ ◡(𝐷 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐷)
5	4	coeq1i 5859	. . . 4 ⊢ (◡(𝐷 ∘ 𝐺) ∘ ◡𝐹) = ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹)
6	2, 3, 5	3eqtri 2764	. . 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 42945	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
11	10	pwexd 5377	. . . . . 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 42847	. . . . 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 42853	. . . . 5 ⊢ (𝜑 → ◡𝐷 = 𝐷)
17	14, 16	coeq12d 5864	. . . 4 ⊢ (𝜑 → (◡𝐺 ∘ ◡𝐷) = (𝐹 ∘ 𝐷))
18	7, 11, 10, 13, 12	fsovcnvd 42847	. . . 4 ⊢ (𝜑 → ◡𝐹 = 𝐺)
19	17, 18	coeq12d 5864	. . 3 ⊢ (𝜑 → ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹) = ((𝐹 ∘ 𝐷) ∘ 𝐺))
20	6, 19	eqtrid 2784	. 2 ⊢ (𝜑 → ◡𝐻 = ((𝐹 ∘ 𝐷) ∘ 𝐺))
21		coass 6264	. . 3 ⊢ ((𝐹 ∘ 𝐷) ∘ 𝐺) = (𝐹 ∘ (𝐷 ∘ 𝐺))
22	21, 1	eqtr4i 2763	. 2 ⊢ ((𝐹 ∘ 𝐷) ∘ 𝐺) = 𝐻
23	20, 22	eqtrdi 2788	1 ⊢ (𝜑 → ◡𝐻 = 𝐻)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474   ∖ cdif 3945  𝒫 cpw 4602   class class class wbr 5148   ↦ cmpt 5231  ^◡ccnv 5675   ∘ ccom 5680  ‘cfv 6543  (class class class)co 7411   ∈ cmpo 7413   ↑_m cmap 8822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824

This theorem is referenced by:  neicvgnvor  42949