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Theorem neifil 23888
Description: The neighborhoods of a nonempty set is a filter. Example 2 of [BourbakiTop1] p. I.36. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
neifil ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ∈ (Fil‘𝑋))

Proof of Theorem neifil
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 toponuni 22920 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
21adantr 480 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 = 𝐽)
3 topontop 22919 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
43adantr 480 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝐽 ∈ Top)
5 simpr 484 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑆𝑋)
65, 2sseqtrd 4020 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑆 𝐽)
7 eqid 2737 . . . . . . . . 9 𝐽 = 𝐽
87neiuni 23130 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝐽 = ((nei‘𝐽)‘𝑆))
94, 6, 8syl2anc 584 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝐽 = ((nei‘𝐽)‘𝑆))
102, 9eqtrd 2777 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 = ((nei‘𝐽)‘𝑆))
11 eqimss2 4043 . . . . . 6 (𝑋 = ((nei‘𝐽)‘𝑆) → ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
1210, 11syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
13 sspwuni 5100 . . . . 5 (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
1412, 13sylibr 234 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
15143adant3 1133 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
16 0nnei 23120 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
173, 16sylan 580 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
18173adant2 1132 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
197tpnei 23129 . . . . . . 7 (𝐽 ∈ Top → (𝑆 𝐽 𝐽 ∈ ((nei‘𝐽)‘𝑆)))
2019biimpa 476 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝐽 ∈ ((nei‘𝐽)‘𝑆))
214, 6, 20syl2anc 584 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝐽 ∈ ((nei‘𝐽)‘𝑆))
222, 21eqeltrd 2841 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
23223adant3 1133 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
2415, 18, 233jca 1129 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
25 elpwi 4607 . . . . 5 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
264ad2antrr 726 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝐽 ∈ Top)
27 simprl 771 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑦 ∈ ((nei‘𝐽)‘𝑆))
28 simprr 773 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑦𝑥)
29 simplr 769 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑥𝑋)
302ad2antrr 726 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑋 = 𝐽)
3129, 30sseqtrd 4020 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑥 𝐽)
327ssnei2 23124 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑦𝑥𝑥 𝐽)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
3326, 27, 28, 31, 32syl22anc 839 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
3433rexlimdvaa 3156 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) → (∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
3525, 34sylan2 593 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
3635ralrimiva 3146 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
37363adant3 1133 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
38 innei 23133 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆))
39383expib 1123 . . . . 5 (𝐽 ∈ Top → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆)))
403, 39syl 17 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆)))
41403ad2ant1 1134 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆)))
4241ralrimivv 3200 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦 ∈ ((nei‘𝐽)‘𝑆)(𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆))
43 isfil2 23864 . 2 (((nei‘𝐽)‘𝑆) ∈ (Fil‘𝑋) ↔ ((((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)) ∧ ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦 ∈ ((nei‘𝐽)‘𝑆)(𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆)))
4424, 37, 42, 43syl3anbrc 1344 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600   cuni 4907  cfv 6561  Topctop 22899  TopOnctopon 22916  neicnei 23105  Filcfil 23853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-fbas 21361  df-top 22900  df-topon 22917  df-nei 23106  df-fil 23854
This theorem is referenced by:  trnei  23900  neiflim  23982  hausflim  23989  flimcf  23990  flimclslem  23992  cnpflf2  24008  cnpflf  24009  fclsfnflim  24035  neipcfilu  24305
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