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Theorem neicvgnvo 38832
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 5404 . . . 4 𝐻 = (𝐹 ∘ (𝐷𝐺))
3 cnvco 5415 . . . 4 (𝐹 ∘ (𝐷𝐺)) = ((𝐷𝐺) ∘ 𝐹)
4 cnvco 5415 . . . . 5 (𝐷𝐺) = (𝐺𝐷)
54coeq1i 5389 . . . 4 ((𝐷𝐺) ∘ 𝐹) = ((𝐺𝐷) ∘ 𝐹)
62, 3, 53eqtri 2750 . . 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 38829 . . . . . 6 (𝜑𝐵 ∈ V)
11 pwexg 4955 . . . . . . 7 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
1210, 11syl 17 . . . . . 6 (𝜑 → 𝒫 𝐵 ∈ V)
13 neicvg.g . . . . . 6 𝐺 = (𝐵𝑂𝒫 𝐵)
14 neicvg.f . . . . . 6 𝐹 = (𝒫 𝐵𝑂𝐵)
157, 10, 12, 13, 14fsovcnvd 38727 . . . . 5 (𝜑𝐺 = 𝐹)
16 neicvg.p . . . . . 6 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
1716, 8, 10dssmapnvod 38733 . . . . 5 (𝜑𝐷 = 𝐷)
1815, 17coeq12d 5394 . . . 4 (𝜑 → (𝐺𝐷) = (𝐹𝐷))
197, 12, 10, 14, 13fsovcnvd 38727 . . . 4 (𝜑𝐹 = 𝐺)
2018, 19coeq12d 5394 . . 3 (𝜑 → ((𝐺𝐷) ∘ 𝐹) = ((𝐹𝐷) ∘ 𝐺))
216, 20syl5eq 2770 . 2 (𝜑𝐻 = ((𝐹𝐷) ∘ 𝐺))
22 coass 5767 . . 3 ((𝐹𝐷) ∘ 𝐺) = (𝐹 ∘ (𝐷𝐺))
2322, 1eqtr4i 2749 . 2 ((𝐹𝐷) ∘ 𝐺) = 𝐻
2421, 23syl6eq 2774 1 (𝜑𝐻 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1596  wcel 2103  {crab 3018  Vcvv 3304  cdif 3677  𝒫 cpw 4266   class class class wbr 4760  cmpt 4837  ccnv 5217  ccom 5222  cfv 6001  (class class class)co 6765  cmpt2 6767  𝑚 cmap 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-map 7976
This theorem is referenced by:  neicvgnvor  38833
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