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Theorem neicvgel1 39023
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 39016 . . 3 (𝜑𝐵 ∈ V)
5 neicvg.o . . . . . 6 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
6 simpr 477 . . . . . . 7 ((𝜑𝐵 ∈ V) → 𝐵 ∈ V)
76pwexd 5015 . . . . . 6 ((𝜑𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
8 neicvg.f . . . . . 6 𝐹 = (𝒫 𝐵𝑂𝐵)
95, 7, 6, 8fsovf1od 38916 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
10 f1ofn 6321 . . . . 5 (𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) → 𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵))
119, 10syl 17 . . . 4 ((𝜑𝐵 ∈ V) → 𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵))
12 neicvg.p . . . . . 6 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
1312, 1, 6dssmapf1od 38921 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
14 f1of 6320 . . . . 5 (𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵) → 𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)⟶(𝒫 𝐵𝑚 𝒫 𝐵))
1513, 14syl 17 . . . 4 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)⟶(𝒫 𝐵𝑚 𝒫 𝐵))
16 neicvg.g . . . . 5 𝐺 = (𝐵𝑂𝒫 𝐵)
175, 6, 7, 16fsovfd 38912 . . . 4 ((𝜑𝐵 ∈ V) → 𝐺:(𝒫 𝒫 𝐵𝑚 𝐵)⟶(𝒫 𝐵𝑚 𝒫 𝐵))
182breqi 4815 . . . . . 6 (𝑁𝐻𝑀𝑁(𝐹 ∘ (𝐷𝐺))𝑀)
193, 18sylib 209 . . . . 5 (𝜑𝑁(𝐹 ∘ (𝐷𝐺))𝑀)
2019adantr 472 . . . 4 ((𝜑𝐵 ∈ V) → 𝑁(𝐹 ∘ (𝐷𝐺))𝑀)
2111, 15, 17, 20brcofffn 38935 . . 3 ((𝜑𝐵 ∈ V) → (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀))
224, 21mpdan 678 . 2 (𝜑 → (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀))
23 simpr2 1250 . . . 4 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)))
24 neicvgel.x . . . . 5 (𝜑𝑋𝐵)
2524adantr 472 . . . 4 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → 𝑋𝐵)
26 neicvgel.s . . . . 5 (𝜑𝑆 ∈ 𝒫 𝐵)
2726adantr 472 . . . 4 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → 𝑆 ∈ 𝒫 𝐵)
2812, 1, 23, 25, 27ntrclselnel1 38961 . . 3 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝑋 ∈ ((𝐺𝑁)‘𝑆) ↔ ¬ 𝑋 ∈ ((𝐷‘(𝐺𝑁))‘(𝐵𝑆))))
29 eqid 2765 . . . 4 (𝒫 𝐵𝑂𝐵) = (𝒫 𝐵𝑂𝐵)
30 simpr1 1248 . . . . 5 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → 𝑁𝐺(𝐺𝑁))
3116breqi 4815 . . . . . . 7 (𝑁𝐺(𝐺𝑁) ↔ 𝑁(𝐵𝑂𝒫 𝐵)(𝐺𝑁))
3231a1i 11 . . . . . 6 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝑁𝐺(𝐺𝑁) ↔ 𝑁(𝐵𝑂𝒫 𝐵)(𝐺𝑁)))
334adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → 𝐵 ∈ V)
34 id 22 . . . . . . . . 9 (𝐵 ∈ V → 𝐵 ∈ V)
35 pwexg 5014 . . . . . . . . 9 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
36 eqid 2765 . . . . . . . . 9 (𝐵𝑂𝒫 𝐵) = (𝐵𝑂𝒫 𝐵)
375, 34, 35, 36fsovf1od 38916 . . . . . . . 8 (𝐵 ∈ V → (𝐵𝑂𝒫 𝐵):(𝒫 𝒫 𝐵𝑚 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
3833, 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 6323 . . . . . . 7 ((𝐵𝑂𝒫 𝐵):(𝒫 𝒫 𝐵𝑚 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵) → Rel (𝐵𝑂𝒫 𝐵))
40 relbrcnvg 5686 . . . . . . 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, 34, 35, 36, 29fsovcnvd 38914 . . . . . . . 8 (𝐵 ∈ V → (𝐵𝑂𝒫 𝐵) = (𝒫 𝐵𝑂𝐵))
4342breqd 4820 . . . . . . 7 (𝐵 ∈ V → ((𝐺𝑁)(𝐵𝑂𝒫 𝐵)𝑁 ↔ (𝐺𝑁)(𝒫 𝐵𝑂𝐵)𝑁))
4433, 43syl 17 . . . . . 6 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → ((𝐺𝑁)(𝐵𝑂𝒫 𝐵)𝑁 ↔ (𝐺𝑁)(𝒫 𝐵𝑂𝐵)𝑁))
4532, 41, 443bitr2d 298 . . . . 5 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝑁𝐺(𝐺𝑁) ↔ (𝐺𝑁)(𝒫 𝐵𝑂𝐵)𝑁))
4630, 45mpbid 223 . . . 4 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝐺𝑁)(𝒫 𝐵𝑂𝐵)𝑁)
475, 29, 46, 25, 27ntrneiel 38985 . . 3 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝑋 ∈ ((𝐺𝑁)‘𝑆) ↔ 𝑆 ∈ (𝑁𝑋)))
48 simpr3 1252 . . . . 5 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝐷‘(𝐺𝑁))𝐹𝑀)
49 difssd 3900 . . . . . . 7 (𝜑 → (𝐵𝑆) ⊆ 𝐵)
504, 49sselpwd 4968 . . . . . 6 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
5150adantr 472 . . . . 5 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝐵𝑆) ∈ 𝒫 𝐵)
525, 8, 48, 25, 51ntrneiel 38985 . . . 4 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝑋 ∈ ((𝐷‘(𝐺𝑁))‘(𝐵𝑆)) ↔ (𝐵𝑆) ∈ (𝑀𝑋)))
5352notbid 309 . . 3 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (¬ 𝑋 ∈ ((𝐷‘(𝐺𝑁))‘(𝐵𝑆)) ↔ ¬ (𝐵𝑆) ∈ (𝑀𝑋)))
5428, 47, 533bitr3d 300 . 2 ((𝜑 ∧ (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀)) → (𝑆 ∈ (𝑁𝑋) ↔ ¬ (𝐵𝑆) ∈ (𝑀𝑋)))
5522, 54mpdan 678 1 (𝜑 → (𝑆 ∈ (𝑁𝑋) ↔ ¬ (𝐵𝑆) ∈ (𝑀𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  {crab 3059  Vcvv 3350  cdif 3729  𝒫 cpw 4315   class class class wbr 4809  cmpt 4888  ccnv 5276  ccom 5281  Rel wrel 5282   Fn wfn 6063  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  cmpt2 6844  𝑚 cmap 8060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-map 8062
This theorem is referenced by:  neicvgel2  39024  neicvgfv  39025
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