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Theorem neifil 23384
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 22416 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
21adantr 482 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
3 topontop 22415 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
43adantr 482 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝐽 ∈ Top)
5 simpr 486 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† 𝑋)
65, 2sseqtrd 4023 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† βˆͺ 𝐽)
7 eqid 2733 . . . . . . . . 9 βˆͺ 𝐽 = βˆͺ 𝐽
87neiuni 22626 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ βˆͺ 𝐽 = βˆͺ ((neiβ€˜π½)β€˜π‘†))
94, 6, 8syl2anc 585 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ βˆͺ 𝐽 = βˆͺ ((neiβ€˜π½)β€˜π‘†))
102, 9eqtrd 2773 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝑋 = βˆͺ ((neiβ€˜π½)β€˜π‘†))
11 eqimss2 4042 . . . . . 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 5104 . . . . 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 1133 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ ((neiβ€˜π½)β€˜π‘†) βŠ† 𝒫 𝑋)
16 0nnei 22616 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 β‰  βˆ…) β†’ Β¬ βˆ… ∈ ((neiβ€˜π½)β€˜π‘†))
173, 16sylan 581 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 β‰  βˆ…) β†’ Β¬ βˆ… ∈ ((neiβ€˜π½)β€˜π‘†))
18173adant2 1132 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ Β¬ βˆ… ∈ ((neiβ€˜π½)β€˜π‘†))
197tpnei 22625 . . . . . . 7 (𝐽 ∈ Top β†’ (𝑆 βŠ† βˆͺ 𝐽 ↔ βˆͺ 𝐽 ∈ ((neiβ€˜π½)β€˜π‘†)))
2019biimpa 478 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ βˆͺ 𝐽 ∈ ((neiβ€˜π½)β€˜π‘†))
214, 6, 20syl2anc 585 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ βˆͺ 𝐽 ∈ ((neiβ€˜π½)β€˜π‘†))
222, 21eqeltrd 2834 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝑋 ∈ ((neiβ€˜π½)β€˜π‘†))
23223adant3 1133 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ 𝑋 ∈ ((neiβ€˜π½)β€˜π‘†))
2415, 18, 233jca 1129 . 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 4610 . . . . 5 (π‘₯ ∈ 𝒫 𝑋 β†’ π‘₯ βŠ† 𝑋)
264ad2antrr 725 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ βŠ† 𝑋) ∧ (𝑦 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 βŠ† π‘₯)) β†’ 𝐽 ∈ Top)
27 simprl 770 . . . . . . 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 772 . . . . . . 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 768 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ βŠ† 𝑋) ∧ (𝑦 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 βŠ† π‘₯)) β†’ π‘₯ βŠ† 𝑋)
302ad2antrr 725 . . . . . . . 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 4023 . . . . . . 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 22620 . . . . . . 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 838 . . . . . 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 3157 . . . . 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 594 . . . 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 3147 . . 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 1133 . 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 22629 . . . . . 6 ((𝐽 ∈ Top ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∩ 𝑦) ∈ ((neiβ€˜π½)β€˜π‘†))
39383expib 1123 . . . . 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 1134 . . 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 3199 . 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 23360 . 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 1344 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22395  TopOnctopon 22412  neicnei 22601  Filcfil 23349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-fbas 20941  df-top 22396  df-topon 22413  df-nei 22602  df-fil 23350
This theorem is referenced by:  trnei  23396  neiflim  23478  hausflim  23485  flimcf  23486  flimclslem  23488  cnpflf2  23504  cnpflf  23505  fclsfnflim  23531  neipcfilu  23801
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