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

Theorem neicvgnvor 38934
 Description: If neighborhood and convergent functions are related by operator 𝐻, the relationship holds with the functions swapped. (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
neicvgnvor (𝜑𝑀𝐻𝑁)
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 neicvgnvor
StepHypRef Expression
1 neicvg.r . . 3 (𝜑𝑁𝐻𝑀)
2 neicvg.o . . . . 5 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
3 neicvg.p . . . . 5 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
4 neicvg.d . . . . 5 𝐷 = (𝑃𝐵)
5 neicvg.f . . . . 5 𝐹 = (𝒫 𝐵𝑂𝐵)
6 neicvg.g . . . . 5 𝐺 = (𝐵𝑂𝒫 𝐵)
7 neicvg.h . . . . 5 𝐻 = (𝐹 ∘ (𝐷𝐺))
82, 3, 4, 5, 6, 7, 1neicvgnvo 38933 . . . 4 (𝜑𝐻 = 𝐻)
98breqd 4815 . . 3 (𝜑 → (𝑁𝐻𝑀𝑁𝐻𝑀))
101, 9mpbird 247 . 2 (𝜑𝑁𝐻𝑀)
11 relco 5794 . . . 4 Rel (𝐹 ∘ (𝐷𝐺))
127releqi 5359 . . . 4 (Rel 𝐻 ↔ Rel (𝐹 ∘ (𝐷𝐺)))
1311, 12mpbir 221 . . 3 Rel 𝐻
1413relbrcnv 5664 . 2 (𝑁𝐻𝑀𝑀𝐻𝑁)
1510, 14sylib 208 1 (𝜑𝑀𝐻𝑁)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  {crab 3054  Vcvv 3340   ∖ cdif 3712  𝒫 cpw 4302   class class class wbr 4804   ↦ cmpt 4881  ◡ccnv 5265   ∘ ccom 5270  Rel wrel 5271  ‘cfv 6049  (class class class)co 6814   ↦ cmpt2 6816   ↑𝑚 cmap 8025 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027 This theorem is referenced by:  neicvgnex  38936
 Copyright terms: Public domain W3C validator