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Theorem neifil 17788
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 16882 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
21adantr 451 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  X  =  U. J )
3 topontop 16881 . . . . . . . . 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 3300 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  S  C_ 
U. J )
7 eqid 2366 . . . . . . . . 9  |-  U. J  =  U. J
87neiuni 17076 . . . . . . . 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 2398 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  X  =  U. ( ( nei `  J ) `  S
) )
11 eqimss2 3317 . . . . . 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 4089 . . . . 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 976 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( ( nei `  J ) `
 S )  C_  ~P X )
16 0nnei 17066 . . . . 5  |-  ( ( J  e.  Top  /\  S  =/=  (/) )  ->  -.  (/) 
e.  ( ( nei `  J ) `  S
) )
173, 16sylan 457 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  =/=  (/) )  ->  -.  (/) 
e.  ( ( nei `  J ) `  S
) )
18173adant2 975 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  -.  (/)  e.  ( ( nei `  J
) `  S )
)
197tpnei 17075 . . . . . . 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 2440 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  X  e.  ( ( nei `  J
) `  S )
)
23223adant3 976 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  X  e.  ( ( nei `  J
) `  S )
)
2415, 18, 233jca 1133 . 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 3722 . . . . 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 3300 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  x  C_  U. J
)
327ssnei2 17070 . . . . . . . 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 1184 . . . . . . 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 2752 . . . . 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 2711 . . 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 976 . 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 17079 . . . . . 6  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S )  /\  y  e.  ( ( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
)
40393expib 1155 . . . . 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 977 . . 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 2719 . 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 17764 . 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 1137 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 935    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   E.wrex 2629    i^i cin 3237    C_ wss 3238   (/)c0 3543   ~Pcpw 3714   U.cuni 3929   ` cfv 5358   Topctop 16848  TopOnctopon 16849   neicnei 17051   Filcfil 17753
This theorem is referenced by:  trnei  17800  neiflim  17882  hausflim  17889  flimcf  17890  flimclslem  17892  cnpflf2  17908  cnpflf  17909  fclsfnflim  17935
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-fbas 16590  df-top 16853  df-topon 16856  df-nei 17052  df-fil 17754
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