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

Theorem neicvgnvo 40600
 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 5718 . . . 4 𝐻 = (𝐹 ∘ (𝐷𝐺))
3 cnvco 5729 . . . 4 (𝐹 ∘ (𝐷𝐺)) = ((𝐷𝐺) ∘ 𝐹)
4 cnvco 5729 . . . . 5 (𝐷𝐺) = (𝐺𝐷)
54coeq1i 5703 . . . 4 ((𝐷𝐺) ∘ 𝐹) = ((𝐺𝐷) ∘ 𝐹)
62, 3, 53eqtri 2848 . . 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 40597 . . . . . 6 (𝜑𝐵 ∈ V)
1110pwexd 5253 . . . . . 6 (𝜑 → 𝒫 𝐵 ∈ V)
12 neicvg.g . . . . . 6 𝐺 = (𝐵𝑂𝒫 𝐵)
13 neicvg.f . . . . . 6 𝐹 = (𝒫 𝐵𝑂𝐵)
147, 10, 11, 12, 13fsovcnvd 40495 . . . . 5 (𝜑𝐺 = 𝐹)
15 neicvg.p . . . . . 6 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
1615, 8, 10dssmapnvod 40501 . . . . 5 (𝜑𝐷 = 𝐷)
1714, 16coeq12d 5708 . . . 4 (𝜑 → (𝐺𝐷) = (𝐹𝐷))
187, 11, 10, 13, 12fsovcnvd 40495 . . . 4 (𝜑𝐹 = 𝐺)
1917, 18coeq12d 5708 . . 3 (𝜑 → ((𝐺𝐷) ∘ 𝐹) = ((𝐹𝐷) ∘ 𝐺))
206, 19syl5eq 2868 . 2 (𝜑𝐻 = ((𝐹𝐷) ∘ 𝐺))
21 coass 6091 . . 3 ((𝐹𝐷) ∘ 𝐺) = (𝐹 ∘ (𝐷𝐺))
2221, 1eqtr4i 2847 . 2 ((𝐹𝐷) ∘ 𝐺) = 𝐻
2320, 22syl6eq 2872 1 (𝜑𝐻 = 𝐻)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  {crab 3130  Vcvv 3471   ∖ cdif 3907  𝒫 cpw 4512   class class class wbr 5039   ↦ cmpt 5119  ◡ccnv 5527   ∘ ccom 5532  ‘cfv 6328  (class class class)co 7130   ∈ cmpo 7132   ↑m cmap 8381 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-map 8383 This theorem is referenced by:  neicvgnvor  40601
 Copyright terms: Public domain W3C validator