Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  neicvgel1 Structured version   Visualization version   GIF version

Theorem neicvgel1 38919
Description: A subset being an element of a neighborhood of a point is equivalent to the complement of that subset not being a element of the convergent of 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
neicvgel1 (𝜑 → (𝑆 ∈ (𝑁𝑋) ↔ ¬ (𝐵𝑆) ∈ (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 neicvgel1
StepHypRef Expression
1 neicvg.d . . . 4 𝐷 = (𝑃𝐵)
2 neicvg.h . . . 4 𝐻 = (𝐹 ∘ (𝐷𝐺))
3 neicvg.r . . . 4 (𝜑𝑁𝐻𝑀)
41, 2, 3neicvgbex 38912 . . 3 (𝜑𝐵 ∈ V)
5 neicvg.o . . . . . 6 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
6 simpr 479 . . . . . . 7 ((𝜑𝐵 ∈ V) → 𝐵 ∈ V)
7 pwexg 4999 . . . . . . 7 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
86, 7syl 17 . . . . . 6 ((𝜑𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
9 neicvg.f . . . . . 6 𝐹 = (𝒫 𝐵𝑂𝐵)
105, 8, 6, 9fsovf1od 38812 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
11 f1ofn 6299 . . . . 5 (𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) → 𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵))
1210, 11syl 17 . . . 4 ((𝜑𝐵 ∈ V) → 𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵))
13 neicvg.p . . . . . 6 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
1413, 1, 6dssmapf1od 38817 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
15 f1of 6298 . . . . 5 (𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵) → 𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)⟶(𝒫 𝐵𝑚 𝒫 𝐵))
1614, 15syl 17 . . . 4 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)⟶(𝒫 𝐵𝑚 𝒫 𝐵))
17 neicvg.g . . . . 5 𝐺 = (𝐵𝑂𝒫 𝐵)
185, 6, 8, 17fsovfd 38808 . . . 4 ((𝜑𝐵 ∈ V) → 𝐺:(𝒫 𝒫 𝐵𝑚 𝐵)⟶(𝒫 𝐵𝑚 𝒫 𝐵))
192breqi 4810 . . . . . 6 (𝑁𝐻𝑀𝑁(𝐹 ∘ (𝐷𝐺))𝑀)
203, 19sylib 208 . . . . 5 (𝜑𝑁(𝐹 ∘ (𝐷𝐺))𝑀)
2120adantr 472 . . . 4 ((𝜑𝐵 ∈ V) → 𝑁(𝐹 ∘ (𝐷𝐺))𝑀)
2212, 16, 18, 21brcofffn 38831 . . 3 ((𝜑𝐵 ∈ V) → (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀))
234, 22mpdan 705 . 2 (𝜑 → (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀))
24 simpr2 1236 . . . 4 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)))
25 neicvgel.x . . . . 5 (𝜑𝑋𝐵)
2625adantr 472 . . . 4 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → 𝑋𝐵)
27 neicvgel.s . . . . 5 (𝜑𝑆 ∈ 𝒫 𝐵)
2827adantr 472 . . . 4 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → 𝑆 ∈ 𝒫 𝐵)
2913, 1, 24, 26, 28ntrclselnel1 38857 . . 3 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝑋 ∈ ((𝐺𝑁)‘𝑆) ↔ ¬ 𝑋 ∈ ((𝐷‘(𝐺𝑁))‘(𝐵𝑆))))
30 eqid 2760 . . . 4 (𝒫 𝐵𝑂𝐵) = (𝒫 𝐵𝑂𝐵)
31 simpr1 1234 . . . . 5 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → 𝑁𝐺(𝐺𝑁))
3217breqi 4810 . . . . . . 7 (𝑁𝐺(𝐺𝑁) ↔ 𝑁(𝐵𝑂𝒫 𝐵)(𝐺𝑁))
3332a1i 11 . . . . . 6 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝑁𝐺(𝐺𝑁) ↔ 𝑁(𝐵𝑂𝒫 𝐵)(𝐺𝑁)))
344adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → 𝐵 ∈ V)
35 id 22 . . . . . . . . 9 (𝐵 ∈ V → 𝐵 ∈ V)
36 eqid 2760 . . . . . . . . 9 (𝐵𝑂𝒫 𝐵) = (𝐵𝑂𝒫 𝐵)
375, 35, 7, 36fsovf1od 38812 . . . . . . . 8 (𝐵 ∈ V → (𝐵𝑂𝒫 𝐵):(𝒫 𝒫 𝐵𝑚 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
3834, 37syl 17 . . . . . . 7 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝐵𝑂𝒫 𝐵):(𝒫 𝒫 𝐵𝑚 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
39 f1orel 6301 . . . . . . 7 ((𝐵𝑂𝒫 𝐵):(𝒫 𝒫 𝐵𝑚 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵) → Rel (𝐵𝑂𝒫 𝐵))
40 relbrcnvg 5662 . . . . . . 7 (Rel (𝐵𝑂𝒫 𝐵) → ((𝐺𝑁)(𝐵𝑂𝒫 𝐵)𝑁𝑁(𝐵𝑂𝒫 𝐵)(𝐺𝑁)))
4138, 39, 403syl 18 . . . . . 6 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → ((𝐺𝑁)(𝐵𝑂𝒫 𝐵)𝑁𝑁(𝐵𝑂𝒫 𝐵)(𝐺𝑁)))
425, 35, 7, 36, 30fsovcnvd 38810 . . . . . . . 8 (𝐵 ∈ V → (𝐵𝑂𝒫 𝐵) = (𝒫 𝐵𝑂𝐵))
4342breqd 4815 . . . . . . 7 (𝐵 ∈ V → ((𝐺𝑁)(𝐵𝑂𝒫 𝐵)𝑁 ↔ (𝐺𝑁)(𝒫 𝐵𝑂𝐵)𝑁))
4434, 43syl 17 . . . . . 6 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → ((𝐺𝑁)(𝐵𝑂𝒫 𝐵)𝑁 ↔ (𝐺𝑁)(𝒫 𝐵𝑂𝐵)𝑁))
4533, 41, 443bitr2d 296 . . . . 5 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝑁𝐺(𝐺𝑁) ↔ (𝐺𝑁)(𝒫 𝐵𝑂𝐵)𝑁))
4631, 45mpbid 222 . . . 4 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝐺𝑁)(𝒫 𝐵𝑂𝐵)𝑁)
475, 30, 46, 26, 28ntrneiel 38881 . . 3 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝑋 ∈ ((𝐺𝑁)‘𝑆) ↔ 𝑆 ∈ (𝑁𝑋)))
48 simpr3 1238 . . . . 5 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝐷‘(𝐺𝑁))𝐹𝑀)
49 difssd 3881 . . . . . . 7 (𝜑 → (𝐵𝑆) ⊆ 𝐵)
504, 49sselpwd 4959 . . . . . 6 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
5150adantr 472 . . . . 5 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝐵𝑆) ∈ 𝒫 𝐵)
525, 9, 48, 26, 51ntrneiel 38881 . . . 4 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝑋 ∈ ((𝐷‘(𝐺𝑁))‘(𝐵𝑆)) ↔ (𝐵𝑆) ∈ (𝑀𝑋)))
5352notbid 307 . . 3 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (¬ 𝑋 ∈ ((𝐷‘(𝐺𝑁))‘(𝐵𝑆)) ↔ ¬ (𝐵𝑆) ∈ (𝑀𝑋)))
5429, 47, 533bitr3d 298 . 2 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝑆 ∈ (𝑁𝑋) ↔ ¬ (𝐵𝑆) ∈ (𝑀𝑋)))
5523, 54mpdan 705 1 (𝜑 → (𝑆 ∈ (𝑁𝑋) ↔ ¬ (𝐵𝑆) ∈ (𝑀𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = 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   Fn wfn 6044  wf 6045  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑚 cmap 8023
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 7114
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 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025
This theorem is referenced by:  neicvgel2  38920  neicvgfv  38921
  Copyright terms: Public domain W3C validator