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Theorem neicvgel2 40348
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 40341 . . 3 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
101, 2, 3, 4, 5, 6, 7, 8, 9neicvgel1 40347 . 2 (𝜑 → ((𝐵𝑆) ∈ (𝑁𝑋) ↔ ¬ (𝐵 ∖ (𝐵𝑆)) ∈ (𝑀𝑋)))
11 neicvgel.s . . . . . 6 (𝜑𝑆 ∈ 𝒫 𝐵)
1211elpwid 4549 . . . . 5 (𝜑𝑆𝐵)
13 dfss4 4232 . . . . 5 (𝑆𝐵 ↔ (𝐵 ∖ (𝐵𝑆)) = 𝑆)
1412, 13sylib 219 . . . 4 (𝜑 → (𝐵 ∖ (𝐵𝑆)) = 𝑆)
1514eleq1d 2894 . . 3 (𝜑 → ((𝐵 ∖ (𝐵𝑆)) ∈ (𝑀𝑋) ↔ 𝑆 ∈ (𝑀𝑋)))
1615notbid 319 . 2 (𝜑 → (¬ (𝐵 ∖ (𝐵𝑆)) ∈ (𝑀𝑋) ↔ ¬ 𝑆 ∈ (𝑀𝑋)))
1710, 16bitrd 280 1 (𝜑 → ((𝐵𝑆) ∈ (𝑁𝑋) ↔ ¬ 𝑆 ∈ (𝑀𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  cdif 3930  wss 3933  𝒫 cpw 4535   class class class wbr 5057  cmpt 5137  ccom 5552  cfv 6348  (class class class)co 7145  cmpo 7147  m cmap 8395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397
This theorem is referenced by: (None)
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