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Theorem neifil 21731
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 20767 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
21adantr 480 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 = 𝐽)
3 topontop 20766 . . . . . . . . 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 3674 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑆 𝐽)
7 eqid 2651 . . . . . . . . 9 𝐽 = 𝐽
87neiuni 20974 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝐽 = ((nei‘𝐽)‘𝑆))
94, 6, 8syl2anc 694 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝐽 = ((nei‘𝐽)‘𝑆))
102, 9eqtrd 2685 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 = ((nei‘𝐽)‘𝑆))
11 eqimss2 3691 . . . . . 6 (𝑋 = ((nei‘𝐽)‘𝑆) → ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
1210, 11syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
13 sspwuni 4643 . . . . 5 (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
1412, 13sylibr 224 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
15143adant3 1101 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
16 0nnei 20964 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
173, 16sylan 487 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
18173adant2 1100 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
197tpnei 20973 . . . . . . 7 (𝐽 ∈ Top → (𝑆 𝐽 𝐽 ∈ ((nei‘𝐽)‘𝑆)))
2019biimpa 500 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝐽 ∈ ((nei‘𝐽)‘𝑆))
214, 6, 20syl2anc 694 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝐽 ∈ ((nei‘𝐽)‘𝑆))
222, 21eqeltrd 2730 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
23223adant3 1101 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
2415, 18, 233jca 1261 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
25 elpwi 4201 . . . . 5 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
264ad2antrr 762 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝐽 ∈ Top)
27 simprl 809 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑦 ∈ ((nei‘𝐽)‘𝑆))
28 simprr 811 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑦𝑥)
29 simplr 807 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑥𝑋)
302ad2antrr 762 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑋 = 𝐽)
3129, 30sseqtrd 3674 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑥 𝐽)
327ssnei2 20968 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑦𝑥𝑥 𝐽)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
3326, 27, 28, 31, 32syl22anc 1367 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
3433rexlimdvaa 3061 . . . . 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 2995 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
37363adant3 1101 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
38 innei 20977 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆))
39383expib 1287 . . . . 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 1102 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆)))
4241ralrimivv 2999 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦 ∈ ((nei‘𝐽)‘𝑆)(𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆))
43 isfil2 21707 . 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 1265 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 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468  cfv 5926  Topctop 20746  TopOnctopon 20763  neicnei 20949  Filcfil 21696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-fbas 19791  df-top 20747  df-topon 20764  df-nei 20950  df-fil 21697
This theorem is referenced by:  trnei  21743  neiflim  21825  hausflim  21832  flimcf  21833  flimclslem  21835  cnpflf2  21851  cnpflf  21852  fclsfnflim  21878  neipcfilu  22147
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