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Theorem neifil 22464
 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 21498 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
21adantr 484 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 = 𝐽)
3 topontop 21497 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
43adantr 484 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝐽 ∈ Top)
5 simpr 488 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑆𝑋)
65, 2sseqtrd 3983 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑆 𝐽)
7 eqid 2821 . . . . . . . . 9 𝐽 = 𝐽
87neiuni 21706 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝐽 = ((nei‘𝐽)‘𝑆))
94, 6, 8syl2anc 587 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝐽 = ((nei‘𝐽)‘𝑆))
102, 9eqtrd 2856 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 = ((nei‘𝐽)‘𝑆))
11 eqimss2 4000 . . . . . 6 (𝑋 = ((nei‘𝐽)‘𝑆) → ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
1210, 11syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
13 sspwuni 4995 . . . . 5 (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
1412, 13sylibr 237 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
15143adant3 1129 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
16 0nnei 21696 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
173, 16sylan 583 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
18173adant2 1128 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
197tpnei 21705 . . . . . . 7 (𝐽 ∈ Top → (𝑆 𝐽 𝐽 ∈ ((nei‘𝐽)‘𝑆)))
2019biimpa 480 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝐽 ∈ ((nei‘𝐽)‘𝑆))
214, 6, 20syl2anc 587 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝐽 ∈ ((nei‘𝐽)‘𝑆))
222, 21eqeltrd 2912 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
23223adant3 1129 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
2415, 18, 233jca 1125 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
25 elpwi 4521 . . . . 5 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
264ad2antrr 725 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝐽 ∈ Top)
27 simprl 770 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑦 ∈ ((nei‘𝐽)‘𝑆))
28 simprr 772 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑦𝑥)
29 simplr 768 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑥𝑋)
302ad2antrr 725 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑋 = 𝐽)
3129, 30sseqtrd 3983 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑥 𝐽)
327ssnei2 21700 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑦𝑥𝑥 𝐽)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
3326, 27, 28, 31, 32syl22anc 837 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
3433rexlimdvaa 3271 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥𝑋) → (∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
3525, 34sylan2 595 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
3635ralrimiva 3170 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
37363adant3 1129 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦𝑥𝑥 ∈ ((nei‘𝐽)‘𝑆)))
38 innei 21709 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆))
39383expib 1119 . . . . 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 1130 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆)))
4241ralrimivv 3178 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦 ∈ ((nei‘𝐽)‘𝑆)(𝑥𝑦) ∈ ((nei‘𝐽)‘𝑆))
43 isfil2 22440 . 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 1340 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ∈ (Fil‘𝑋))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ≠ wne 3007  ∀wral 3126  ∃wrex 3127   ∩ cin 3909   ⊆ wss 3910  ∅c0 4266  𝒫 cpw 4512  ∪ cuni 4811  ‘cfv 6328  Topctop 21477  TopOnctopon 21494  neicnei 21681  Filcfil 22429 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-fbas 20518  df-top 21478  df-topon 21495  df-nei 21682  df-fil 22430 This theorem is referenced by:  trnei  22476  neiflim  22558  hausflim  22565  flimcf  22566  flimclslem  22568  cnpflf2  22584  cnpflf  22585  fclsfnflim  22611  neipcfilu  22881
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