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Theorem neifil 21494
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 20542 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
21adantr 480 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 = 𝐽)
3 topontop 20541 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
43adantr 480 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝐽 ∈ Top)
5 simpr 476 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑆𝑋)
65, 2sseqtrd 3604 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑆 𝐽)
7 eqid 2610 . . . . . . . . 9 𝐽 = 𝐽
87neiuni 20736 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝐽 = ((nei‘𝐽)‘𝑆))
94, 6, 8syl2anc 691 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝐽 = ((nei‘𝐽)‘𝑆))
102, 9eqtrd 2644 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 = ((nei‘𝐽)‘𝑆))
11 eqimss2 3621 . . . . . 6 (𝑋 = ((nei‘𝐽)‘𝑆) → ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
1210, 11syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
13 sspwuni 4547 . . . . 5 (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
1412, 13sylibr 223 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
15143adant3 1074 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
16 0nnei 20726 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
173, 16sylan 487 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
18173adant2 1073 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
197tpnei 20735 . . . . . . 7 (𝐽 ∈ Top → (𝑆 𝐽 𝐽 ∈ ((nei‘𝐽)‘𝑆)))
2019biimpa 500 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝐽 ∈ ((nei‘𝐽)‘𝑆))
214, 6, 20syl2anc 691 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝐽 ∈ ((nei‘𝐽)‘𝑆))
222, 21eqeltrd 2688 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
23223adant3 1074 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
2415, 18, 233jca 1235 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
25 elpwi 4117 . . . . 5 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
264ad2antrr 758 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝐽 ∈ Top)
27 simprl 790 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑦 ∈ ((nei‘𝐽)‘𝑆))
28 simprr 792 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑦𝑥)
29 simplr 788 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑥𝑋)
302ad2antrr 758 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑋 = 𝐽)
3129, 30sseqtrd 3604 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑥 𝐽)
327ssnei2 20730 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑦𝑥𝑥 𝐽)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
3326, 27, 28, 31, 32syl22anc 1319 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
3433rexlimdvaa 3014 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) → (∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
3525, 34sylan2 490 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
3635ralrimiva 2949 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
37363adant3 1074 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
38 innei 20739 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆))
39383expib 1260 . . . . 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 1075 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆)))
4241ralrimivv 2953 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦 ∈ ((nei‘𝐽)‘𝑆)(𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆))
43 isfil2 21470 . 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 1239 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897  cin 3539  wss 3540  c0 3874  𝒫 cpw 4108   cuni 4372  cfv 5804  Topctop 20517  TopOnctopon 20518  neicnei 20711  Filcfil 21459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-fbas 19564  df-top 20521  df-topon 20523  df-nei 20712  df-fil 21460
This theorem is referenced by:  trnei  21506  neiflim  21588  hausflim  21595  flimcf  21596  flimclslem  21598  cnpflf2  21614  cnpflf  21615  fclsfnflim  21641  neipcfilu  21910
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