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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 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
neicvg.p 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
neicvg.d 𝐷 = (𝑃𝐵)
neicvg.f 𝐹 = (𝒫 𝐵𝑂𝐵)
neicvg.g 𝐺 = (𝐵𝑂𝒫 𝐵)
neicvg.h 𝐻 = (𝐹 ∘ (𝐷𝐺))
neicvg.r (𝜑𝑁𝐻𝑀)
Assertion
Ref Expression
neicvgnvo (𝜑𝐻 = 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
StepHypRef Expression
1 neicvg.h . . . . 5 𝐻 = (𝐹 ∘ (𝐷𝐺))
21cnveqi 5498 . . . 4 𝐻 = (𝐹 ∘ (𝐷𝐺))
3 cnvco 5509 . . . 4 (𝐹 ∘ (𝐷𝐺)) = ((𝐷𝐺) ∘ 𝐹)
4 cnvco 5509 . . . . 5 (𝐷𝐺) = (𝐺𝐷)
54coeq1i 5483 . . . 4 ((𝐷𝐺) ∘ 𝐹) = ((𝐺𝐷) ∘ 𝐹)
62, 3, 53eqtri 2823 . . 3 𝐻 = ((𝐺𝐷) ∘ 𝐹)
7 neicvg.o . . . . . 6 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
8 neicvg.d . . . . . . 7 𝐷 = (𝑃𝐵)
9 neicvg.r . . . . . . 7 (𝜑𝑁𝐻𝑀)
108, 1, 9neicvgbex 39179 . . . . . 6 (𝜑𝐵 ∈ V)
1110pwexd 5047 . . . . . 6 (𝜑 → 𝒫 𝐵 ∈ V)
12 neicvg.g . . . . . 6 𝐺 = (𝐵𝑂𝒫 𝐵)
13 neicvg.f . . . . . 6 𝐹 = (𝒫 𝐵𝑂𝐵)
147, 10, 11, 12, 13fsovcnvd 39077 . . . . 5 (𝜑𝐺 = 𝐹)
15 neicvg.p . . . . . 6 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
1615, 8, 10dssmapnvod 39083 . . . . 5 (𝜑𝐷 = 𝐷)
1714, 16coeq12d 5488 . . . 4 (𝜑 → (𝐺𝐷) = (𝐹𝐷))
187, 11, 10, 13, 12fsovcnvd 39077 . . . 4 (𝜑𝐹 = 𝐺)
1917, 18coeq12d 5488 . . 3 (𝜑 → ((𝐺𝐷) ∘ 𝐹) = ((𝐹𝐷) ∘ 𝐺))
206, 19syl5eq 2843 . 2 (𝜑𝐻 = ((𝐹𝐷) ∘ 𝐺))
21 coass 5871 . . 3 ((𝐹𝐷) ∘ 𝐺) = (𝐹 ∘ (𝐷𝐺))
2221, 1eqtr4i 2822 . 2 ((𝐹𝐷) ∘ 𝐺) = 𝐻
2320, 22syl6eq 2847 1 (𝜑𝐻 = 𝐻)
Colors of variables: wff setvar class
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
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