Theorem neicvgnvo 41255

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 5711	. . . 4 ⊢ ◡𝐻 = ◡(𝐹 ∘ (𝐷 ∘ 𝐺))
3		cnvco 5722	. . . 4 ⊢ ◡(𝐹 ∘ (𝐷 ∘ 𝐺)) = (◡(𝐷 ∘ 𝐺) ∘ ◡𝐹)
4		cnvco 5722	. . . . 5 ⊢ ◡(𝐷 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐷)
5	4	coeq1i 5696	. . . 4 ⊢ (◡(𝐷 ∘ 𝐺) ∘ ◡𝐹) = ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹)
6	2, 3, 5	3eqtri 2765	. . 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 41252	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
11	10	pwexd 5243	. . . . . 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 41152	. . . . 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 41158	. . . . 5 ⊢ (𝜑 → ◡𝐷 = 𝐷)
17	14, 16	coeq12d 5701	. . . 4 ⊢ (𝜑 → (◡𝐺 ∘ ◡𝐷) = (𝐹 ∘ 𝐷))
18	7, 11, 10, 13, 12	fsovcnvd 41152	. . . 4 ⊢ (𝜑 → ◡𝐹 = 𝐺)
19	17, 18	coeq12d 5701	. . 3 ⊢ (𝜑 → ((◡𝐺 ∘ ◡𝐷) ∘ ◡𝐹) = ((𝐹 ∘ 𝐷) ∘ 𝐺))
20	6, 19	syl5eq 2785	. 2 ⊢ (𝜑 → ◡𝐻 = ((𝐹 ∘ 𝐷) ∘ 𝐺))
21		coass 6092	. . 3 ⊢ ((𝐹 ∘ 𝐷) ∘ 𝐺) = (𝐹 ∘ (𝐷 ∘ 𝐺))
22	21, 1	eqtr4i 2764	. 2 ⊢ ((𝐹 ∘ 𝐷) ∘ 𝐺) = 𝐻
23	20, 22	eqtrdi 2789	1 ⊢ (𝜑 → ◡𝐻 = 𝐻)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2113  {crab 3057  Vcvv 3397   ∖ cdif 3838  𝒫 cpw 4485   class class class wbr 5027   ↦ cmpt 5107  ^◡ccnv 5518   ∘ ccom 5523  ‘cfv 6333  (class class class)co 7164   ∈ cmpo 7166   ↑_m cmap 8430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-1st 7707  df-2nd 7708  df-map 8432

This theorem is referenced by:  neicvgnvor  41256