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Theorem neifil 23283
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 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ ((neiβ€˜π½)β€˜π‘†) ∈ (Filβ€˜π‘‹))
Proof of Theorem neifil
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 toponuni 22315 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
21adantr 481 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
3 topontop 22314 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
43adantr 481 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝐽 ∈ Top)
5 simpr 485 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† 𝑋)
65, 2sseqtrd 4002 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† βˆͺ 𝐽)
7 eqid 2731 . . . . . . . . 9 βˆͺ 𝐽 = βˆͺ 𝐽
87neiuni 22525 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ βˆͺ 𝐽 = βˆͺ ((neiβ€˜π½)β€˜π‘†))
94, 6, 8syl2anc 584 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ βˆͺ 𝐽 = βˆͺ ((neiβ€˜π½)β€˜π‘†))
102, 9eqtrd 2771 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝑋 = βˆͺ ((neiβ€˜π½)β€˜π‘†))
11 eqimss2 4021 . . . . . 6 (𝑋 = βˆͺ ((neiβ€˜π½)β€˜π‘†) β†’ βˆͺ ((neiβ€˜π½)β€˜π‘†) βŠ† 𝑋)
1210, 11syl 17 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ βˆͺ ((neiβ€˜π½)β€˜π‘†) βŠ† 𝑋)
13 sspwuni 5080 . . . . 5 (((neiβ€˜π½)β€˜π‘†) βŠ† 𝒫 𝑋 ↔ βˆͺ ((neiβ€˜π½)β€˜π‘†) βŠ† 𝑋)
1412, 13sylibr 233 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ ((neiβ€˜π½)β€˜π‘†) βŠ† 𝒫 𝑋)
15143adant3 1132 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ ((neiβ€˜π½)β€˜π‘†) βŠ† 𝒫 𝑋)
16 0nnei 22515 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 β‰  βˆ…) β†’ Β¬ βˆ… ∈ ((neiβ€˜π½)β€˜π‘†))
173, 16sylan 580 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 β‰  βˆ…) β†’ Β¬ βˆ… ∈ ((neiβ€˜π½)β€˜π‘†))
18173adant2 1131 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ Β¬ βˆ… ∈ ((neiβ€˜π½)β€˜π‘†))
197tpnei 22524 . . . . . . 7 (𝐽 ∈ Top β†’ (𝑆 βŠ† βˆͺ 𝐽 ↔ βˆͺ 𝐽 ∈ ((neiβ€˜π½)β€˜π‘†)))
2019biimpa 477 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ βˆͺ 𝐽 ∈ ((neiβ€˜π½)β€˜π‘†))
214, 6, 20syl2anc 584 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ βˆͺ 𝐽 ∈ ((neiβ€˜π½)β€˜π‘†))
222, 21eqeltrd 2832 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝑋 ∈ ((neiβ€˜π½)β€˜π‘†))
23223adant3 1132 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ 𝑋 ∈ ((neiβ€˜π½)β€˜π‘†))
2415, 18, 233jca 1128 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (((neiβ€˜π½)β€˜π‘†) βŠ† 𝒫 𝑋 ∧ Β¬ βˆ… ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑋 ∈ ((neiβ€˜π½)β€˜π‘†)))
25 elpwi 4587 . . . . 5 (π‘₯ ∈ 𝒫 𝑋 β†’ π‘₯ βŠ† 𝑋)
264ad2antrr 724 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ βŠ† 𝑋) ∧ (𝑦 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 βŠ† π‘₯)) β†’ 𝐽 ∈ Top)
27 simprl 769 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ βŠ† 𝑋) ∧ (𝑦 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 βŠ† π‘₯)) β†’ 𝑦 ∈ ((neiβ€˜π½)β€˜π‘†))
28 simprr 771 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ βŠ† 𝑋) ∧ (𝑦 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 βŠ† π‘₯)) β†’ 𝑦 βŠ† π‘₯)
29 simplr 767 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ βŠ† 𝑋) ∧ (𝑦 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 βŠ† π‘₯)) β†’ π‘₯ βŠ† 𝑋)
302ad2antrr 724 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ βŠ† 𝑋) ∧ (𝑦 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 βŠ† π‘₯)) β†’ 𝑋 = βˆͺ 𝐽)
3129, 30sseqtrd 4002 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ βŠ† 𝑋) ∧ (𝑦 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 βŠ† π‘₯)) β†’ π‘₯ βŠ† βˆͺ 𝐽)
327ssnei2 22519 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑦 βŠ† π‘₯ ∧ π‘₯ βŠ† βˆͺ 𝐽)) β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))
3326, 27, 28, 31, 32syl22anc 837 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ βŠ† 𝑋) ∧ (𝑦 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 βŠ† π‘₯)) β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))
3433rexlimdvaa 3155 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ βŠ† 𝑋) β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π‘†)𝑦 βŠ† π‘₯ β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)))
3525, 34sylan2 593 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ 𝒫 𝑋) β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π‘†)𝑦 βŠ† π‘₯ β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)))
3635ralrimiva 3145 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ βˆ€π‘₯ ∈ 𝒫 𝑋(βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π‘†)𝑦 βŠ† π‘₯ β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)))
37363adant3 1132 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ βˆ€π‘₯ ∈ 𝒫 𝑋(βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π‘†)𝑦 βŠ† π‘₯ β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)))
38 innei 22528 . . . . . 6 ((𝐽 ∈ Top ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∩ 𝑦) ∈ ((neiβ€˜π½)β€˜π‘†))
39383expib 1122 . . . . 5 (𝐽 ∈ Top β†’ ((π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∩ 𝑦) ∈ ((neiβ€˜π½)β€˜π‘†)))
403, 39syl 17 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ((π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∩ 𝑦) ∈ ((neiβ€˜π½)β€˜π‘†)))
41403ad2ant1 1133 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ ((π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∩ 𝑦) ∈ ((neiβ€˜π½)β€˜π‘†)))
4241ralrimivv 3197 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ βˆ€π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜π‘†)(π‘₯ ∩ 𝑦) ∈ ((neiβ€˜π½)β€˜π‘†))
43 isfil2 23259 . 2 (((neiβ€˜π½)β€˜π‘†) ∈ (Filβ€˜π‘‹) ↔ ((((neiβ€˜π½)β€˜π‘†) βŠ† 𝒫 𝑋 ∧ Β¬ βˆ… ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑋 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ βˆ€π‘₯ ∈ 𝒫 𝑋(βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π‘†)𝑦 βŠ† π‘₯ β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ βˆ€π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜π‘†)(π‘₯ ∩ 𝑦) ∈ ((neiβ€˜π½)β€˜π‘†)))
4424, 37, 42, 43syl3anbrc 1343 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ ((neiβ€˜π½)β€˜π‘†) ∈ (Filβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2939  βˆ€wral 3060  βˆƒwrex 3069   ∩ cin 3927   βŠ† wss 3928  βˆ…c0 4302  π’« cpw 4580  βˆͺ cuni 4885  β€˜cfv 6516  Topctop 22294  TopOnctopon 22311  neicnei 22500  Filcfil 23248
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-fbas 20845  df-top 22295  df-topon 22312  df-nei 22501  df-fil 23249
This theorem is referenced by:  trnei  23295  neiflim  23377  hausflim  23384  flimcf  23385  flimclslem  23387  cnpflf2  23403  cnpflf  23404  fclsfnflim  23430  neipcfilu  23700
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