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Theorem neifil 23904
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 22936 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
21adantr 480 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 = 𝐽)
3 topontop 22935 . . . . . . . . 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 4036 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑆 𝐽)
7 eqid 2735 . . . . . . . . 9 𝐽 = 𝐽
87neiuni 23146 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝐽 = ((nei‘𝐽)‘𝑆))
94, 6, 8syl2anc 584 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝐽 = ((nei‘𝐽)‘𝑆))
102, 9eqtrd 2775 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 = ((nei‘𝐽)‘𝑆))
11 eqimss2 4055 . . . . . 6 (𝑋 = ((nei‘𝐽)‘𝑆) → ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
1210, 11syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
13 sspwuni 5105 . . . . 5 (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
1412, 13sylibr 234 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
15143adant3 1131 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
16 0nnei 23136 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
173, 16sylan 580 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
18173adant2 1130 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
197tpnei 23145 . . . . . . 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 2839 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
23223adant3 1131 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
2415, 18, 233jca 1127 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
25 elpwi 4612 . . . . 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 4036 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑥 𝐽)
327ssnei2 23140 . . . . . . 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 3154 . . . . 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 3144 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
37363adant3 1131 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
38 innei 23149 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆))
39383expib 1121 . . . . 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 1132 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆)))
4241ralrimivv 3198 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦 ∈ ((nei‘𝐽)‘𝑆)(𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆))
43 isfil2 23880 . 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 1342 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 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912  cfv 6563  Topctop 22915  TopOnctopon 22932  neicnei 23121  Filcfil 23869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-fbas 21379  df-top 22916  df-topon 22933  df-nei 23122  df-fil 23870
This theorem is referenced by:  trnei  23916  neiflim  23998  hausflim  24005  flimcf  24006  flimclslem  24008  cnpflf2  24024  cnpflf  24025  fclsfnflim  24051  neipcfilu  24321
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