Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  neicvgel2 Structured version   Visualization version   GIF version

Theorem neicvgel2 41415
Description: The complement of a subset being an element of a neighborhood at a point is equivalent to that subset not being a element of the convergent at that point. (Contributed by RP, 12-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)
neicvgel.x (𝜑𝑋𝐵)
neicvgel.s (𝜑𝑆 ∈ 𝒫 𝐵)
Assertion
Ref Expression
neicvgel2 (𝜑 → ((𝐵𝑆) ∈ (𝑁𝑋) ↔ ¬ 𝑆 ∈ (𝑀𝑋)))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚   𝐵,𝑛,𝑜,𝑝   𝐷,𝑖,𝑗,𝑘,𝑙,𝑚   𝐷,𝑛,𝑜,𝑝   𝑖,𝐹,𝑗,𝑘,𝑙   𝑛,𝐹,𝑜,𝑝   𝑖,𝐺,𝑗,𝑘,𝑙,𝑚   𝑛,𝐺,𝑜,𝑝   𝑖,𝑀,𝑗,𝑘,𝑙   𝑛,𝑀,𝑜,𝑝   𝑖,𝑁,𝑗,𝑘,𝑙,𝑚   𝑛,𝑁,𝑜,𝑝   𝑆,𝑚   𝑆,𝑜   𝑋,𝑙,𝑚   𝜑,𝑖,𝑗,𝑘,𝑙   𝜑,𝑛,𝑜,𝑝
Allowed substitution hints:   𝜑(𝑚)   𝑃(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑆(𝑖,𝑗,𝑘,𝑛,𝑝,𝑙)   𝐹(𝑚)   𝐻(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑀(𝑚)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑋(𝑖,𝑗,𝑘,𝑛,𝑜,𝑝)

Proof of Theorem neicvgel2
StepHypRef Expression
1 neicvg.o . . 3 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
2 neicvg.p . . 3 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
3 neicvg.d . . 3 𝐷 = (𝑃𝐵)
4 neicvg.f . . 3 𝐹 = (𝒫 𝐵𝑂𝐵)
5 neicvg.g . . 3 𝐺 = (𝐵𝑂𝒫 𝐵)
6 neicvg.h . . 3 𝐻 = (𝐹 ∘ (𝐷𝐺))
7 neicvg.r . . 3 (𝜑𝑁𝐻𝑀)
8 neicvgel.x . . 3 (𝜑𝑋𝐵)
93, 6, 7neicvgrcomplex 41408 . . 3 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
101, 2, 3, 4, 5, 6, 7, 8, 9neicvgel1 41414 . 2 (𝜑 → ((𝐵𝑆) ∈ (𝑁𝑋) ↔ ¬ (𝐵 ∖ (𝐵𝑆)) ∈ (𝑀𝑋)))
11 neicvgel.s . . . . . 6 (𝜑𝑆 ∈ 𝒫 𝐵)
1211elpwid 4529 . . . . 5 (𝜑𝑆𝐵)
13 dfss4 4178 . . . . 5 (𝑆𝐵 ↔ (𝐵 ∖ (𝐵𝑆)) = 𝑆)
1412, 13sylib 221 . . . 4 (𝜑 → (𝐵 ∖ (𝐵𝑆)) = 𝑆)
1514eleq1d 2822 . . 3 (𝜑 → ((𝐵 ∖ (𝐵𝑆)) ∈ (𝑀𝑋) ↔ 𝑆 ∈ (𝑀𝑋)))
1615notbid 321 . 2 (𝜑 → (¬ (𝐵 ∖ (𝐵𝑆)) ∈ (𝑀𝑋) ↔ ¬ 𝑆 ∈ (𝑀𝑋)))
1710, 16bitrd 282 1 (𝜑 → ((𝐵𝑆) ∈ (𝑁𝑋) ↔ ¬ 𝑆 ∈ (𝑀𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1543  wcel 2110  {crab 3065  Vcvv 3413  cdif 3868  wss 3871  𝒫 cpw 4518   class class class wbr 5058  cmpt 5140  ccom 5560  cfv 6385  (class class class)co 7218  cmpo 7220  m cmap 8513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5184  ax-sep 5197  ax-nul 5204  ax-pow 5263  ax-pr 5327  ax-un 7528
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3415  df-sbc 3700  df-csb 3817  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-op 4553  df-uni 4825  df-iun 4911  df-br 5059  df-opab 5121  df-mpt 5141  df-id 5460  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567  df-res 5568  df-ima 5569  df-iota 6343  df-fun 6387  df-fn 6388  df-f 6389  df-f1 6390  df-fo 6391  df-f1o 6392  df-fv 6393  df-ov 7221  df-oprab 7222  df-mpo 7223  df-1st 7766  df-2nd 7767  df-map 8515
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator