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Theorem neifil 17577
Description: The neighborhoods of a non-empty 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  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( ( nei `  J ) `
 S )  e.  ( Fil `  X
) )

Proof of Theorem neifil
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 toponuni 16667 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
21adantr 451 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  X  =  U. J )
3 topontop 16666 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
43adantr 451 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  J  e.  Top )
5 simpr 447 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  S  C_  X )
65, 2sseqtrd 3216 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  S  C_ 
U. J )
7 eqid 2285 . . . . . . . . 9  |-  U. J  =  U. J
87neiuni 16861 . . . . . . . 8  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  U. J  =  U. ( ( nei `  J
) `  S )
)
94, 6, 8syl2anc 642 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  U. J  =  U. ( ( nei `  J ) `  S
) )
102, 9eqtrd 2317 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  X  =  U. ( ( nei `  J ) `  S
) )
11 eqimss2 3233 . . . . . 6  |-  ( X  =  U. ( ( nei `  J ) `
 S )  ->  U. ( ( nei `  J
) `  S )  C_  X )
1210, 11syl 15 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  U. (
( nei `  J
) `  S )  C_  X )
13 sspwuni 3989 . . . . 5  |-  ( ( ( nei `  J
) `  S )  C_ 
~P X  <->  U. (
( nei `  J
) `  S )  C_  X )
1412, 13sylibr 203 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  (
( nei `  J
) `  S )  C_ 
~P X )
15143adant3 975 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( ( nei `  J ) `
 S )  C_  ~P X )
16 0nnei 16851 . . . . 5  |-  ( ( J  e.  Top  /\  S  =/=  (/) )  ->  -.  (/) 
e.  ( ( nei `  J ) `  S
) )
173, 16sylan 457 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  =/=  (/) )  ->  -.  (/) 
e.  ( ( nei `  J ) `  S
) )
18173adant2 974 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  -.  (/)  e.  ( ( nei `  J
) `  S )
)
197tpnei 16860 . . . . . . 7  |-  ( J  e.  Top  ->  ( S  C_  U. J  <->  U. J  e.  ( ( nei `  J
) `  S )
) )
2019biimpa 470 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  U. J  e.  ( ( nei `  J
) `  S )
)
214, 6, 20syl2anc 642 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  U. J  e.  ( ( nei `  J
) `  S )
)
222, 21eqeltrd 2359 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  X  e.  ( ( nei `  J
) `  S )
)
23223adant3 975 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  X  e.  ( ( nei `  J
) `  S )
)
2415, 18, 233jca 1132 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( ( ( nei `  J
) `  S )  C_ 
~P X  /\  -.  (/) 
e.  ( ( nei `  J ) `  S
)  /\  X  e.  ( ( nei `  J
) `  S )
) )
25 elpwi 3635 . . . . 5  |-  ( x  e.  ~P X  ->  x  C_  X )
264ad2antrr 706 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  J  e.  Top )
27 simprl 732 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  y  e.  ( ( nei `  J
) `  S )
)
28 simprr 733 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  y  C_  x )
29 simplr 731 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  x  C_  X
)
302ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  X  =  U. J )
3129, 30sseqtrd 3216 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  x  C_  U. J
)
327ssnei2 16855 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  y  e.  ( ( nei `  J ) `
 S ) )  /\  ( y  C_  x  /\  x  C_  U. J
) )  ->  x  e.  ( ( nei `  J
) `  S )
)
3326, 27, 28, 31, 32syl22anc 1183 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  x  e.  ( ( nei `  J
) `  S )
)
3433expr 598 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  y  e.  ( ( nei `  J
) `  S )
)  ->  ( y  C_  x  ->  x  e.  ( ( nei `  J
) `  S )
) )
3534rexlimdva 2669 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X )  ->  ( E. y  e.  (
( nei `  J
) `  S )
y  C_  x  ->  x  e.  ( ( nei `  J ) `  S
) ) )
3625, 35sylan2 460 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  e.  ~P X )  -> 
( E. y  e.  ( ( nei `  J
) `  S )
y  C_  x  ->  x  e.  ( ( nei `  J ) `  S
) ) )
3736ralrimiva 2628 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  A. x  e.  ~P  X ( E. y  e.  ( ( nei `  J ) `
 S ) y 
C_  x  ->  x  e.  ( ( nei `  J
) `  S )
) )
38373adant3 975 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  A. x  e.  ~P  X ( E. y  e.  ( ( nei `  J ) `
 S ) y 
C_  x  ->  x  e.  ( ( nei `  J
) `  S )
) )
39 innei 16864 . . . . . 6  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S )  /\  y  e.  ( ( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
)
40393expib 1154 . . . . 5  |-  ( J  e.  Top  ->  (
( x  e.  ( ( nei `  J
) `  S )  /\  y  e.  (
( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
) )
413, 40syl 15 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( (
x  e.  ( ( nei `  J ) `
 S )  /\  y  e.  ( ( nei `  J ) `  S ) )  -> 
( x  i^i  y
)  e.  ( ( nei `  J ) `
 S ) ) )
42413ad2ant1 976 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( ( x  e.  ( ( nei `  J ) `
 S )  /\  y  e.  ( ( nei `  J ) `  S ) )  -> 
( x  i^i  y
)  e.  ( ( nei `  J ) `
 S ) ) )
4342ralrimivv 2636 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  A. x  e.  ( ( nei `  J
) `  S ) A. y  e.  (
( nei `  J
) `  S )
( x  i^i  y
)  e.  ( ( nei `  J ) `
 S ) )
44 isfil2 17553 . 2  |-  ( ( ( nei `  J
) `  S )  e.  ( Fil `  X
)  <->  ( ( ( ( nei `  J
) `  S )  C_ 
~P X  /\  -.  (/) 
e.  ( ( nei `  J ) `  S
)  /\  X  e.  ( ( nei `  J
) `  S )
)  /\  A. x  e.  ~P  X ( E. y  e.  ( ( nei `  J ) `
 S ) y 
C_  x  ->  x  e.  ( ( nei `  J
) `  S )
)  /\  A. x  e.  ( ( nei `  J
) `  S ) A. y  e.  (
( nei `  J
) `  S )
( x  i^i  y
)  e.  ( ( nei `  J ) `
 S ) ) )
4524, 38, 43, 44syl3anbrc 1136 1  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( ( nei `  J ) `
 S )  e.  ( Fil `  X
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   E.wrex 2546    i^i cin 3153    C_ wss 3154   (/)c0 3457   ~Pcpw 3627   U.cuni 3829   ` cfv 5257   Topctop 16633  TopOnctopon 16634   neicnei 16836   Filcfil 17542
This theorem is referenced by:  trnei  17589  neiflim  17671  hausflim  17678  flimcf  17679  flimclslem  17681  cnpflf2  17697  cnpflf  17698  fclsfnflim  17724  conttnf2  25573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-top 16638  df-topon 16641  df-nei 16837  df-fbas 17522  df-fil 17543
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