Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  neicvgnvo Structured version   Visualization version   GIF version

Theorem neicvgnvo 42866
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 5875 . . . 4 𝐻 = (𝐹 ∘ (𝐷𝐺))
3 cnvco 5886 . . . 4 (𝐹 ∘ (𝐷𝐺)) = ((𝐷𝐺) ∘ 𝐹)
4 cnvco 5886 . . . . 5 (𝐷𝐺) = (𝐺𝐷)
54coeq1i 5860 . . . 4 ((𝐷𝐺) ∘ 𝐹) = ((𝐺𝐷) ∘ 𝐹)
62, 3, 53eqtri 2765 . . 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 42863 . . . . . 6 (𝜑𝐵 ∈ V)
1110pwexd 5378 . . . . . 6 (𝜑 → 𝒫 𝐵 ∈ V)
12 neicvg.g . . . . . 6 𝐺 = (𝐵𝑂𝒫 𝐵)
13 neicvg.f . . . . . 6 𝐹 = (𝒫 𝐵𝑂𝐵)
147, 10, 11, 12, 13fsovcnvd 42765 . . . . 5 (𝜑𝐺 = 𝐹)
15 neicvg.p . . . . . 6 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
1615, 8, 10dssmapnvod 42771 . . . . 5 (𝜑𝐷 = 𝐷)
1714, 16coeq12d 5865 . . . 4 (𝜑 → (𝐺𝐷) = (𝐹𝐷))
187, 11, 10, 13, 12fsovcnvd 42765 . . . 4 (𝜑𝐹 = 𝐺)
1917, 18coeq12d 5865 . . 3 (𝜑 → ((𝐺𝐷) ∘ 𝐹) = ((𝐹𝐷) ∘ 𝐺))
206, 19eqtrid 2785 . 2 (𝜑𝐻 = ((𝐹𝐷) ∘ 𝐺))
21 coass 6265 . . 3 ((𝐹𝐷) ∘ 𝐺) = (𝐹 ∘ (𝐷𝐺))
2221, 1eqtr4i 2764 . 2 ((𝐹𝐷) ∘ 𝐺) = 𝐻
2320, 22eqtrdi 2789 1 (𝜑𝐻 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  cdif 3946  𝒫 cpw 4603   class class class wbr 5149  cmpt 5232  ccnv 5676  ccom 5681  cfv 6544  (class class class)co 7409  cmpo 7411  m cmap 8820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822
This theorem is referenced by:  neicvgnvor  42867
  Copyright terms: Public domain W3C validator