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Theorem neifil 17407
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
StepHypRef Expression
1 toponuni 16497 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
21adantr 453 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  X  =  U. J )
3 topontop 16496 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
43adantr 453 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  J  e.  Top )
5 simpr 449 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  S  C_  X )
65, 2sseqtrd 3135 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  S  C_ 
U. J )
7 eqid 2253 . . . . . . . . 9  |-  U. J  =  U. J
87neiuni 16691 . . . . . . . 8  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  U. J  =  U. ( ( nei `  J
) `  S )
)
94, 6, 8syl2anc 645 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  U. J  =  U. ( ( nei `  J ) `  S
) )
102, 9eqtrd 2285 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  X  =  U. ( ( nei `  J ) `  S
) )
11 eqimss2 3152 . . . . . 6  |-  ( X  =  U. ( ( nei `  J ) `
 S )  ->  U. ( ( nei `  J
) `  S )  C_  X )
1210, 11syl 17 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  U. (
( nei `  J
) `  S )  C_  X )
13 sspwuni 3885 . . . . 5  |-  ( ( ( nei `  J
) `  S )  C_ 
~P X  <->  U. (
( nei `  J
) `  S )  C_  X )
1412, 13sylibr 205 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  (
( nei `  J
) `  S )  C_ 
~P X )
15143adant3 980 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( ( nei `  J ) `
 S )  C_  ~P X )
16 0nnei 16681 . . . . 5  |-  ( ( J  e.  Top  /\  S  =/=  (/) )  ->  -.  (/) 
e.  ( ( nei `  J ) `  S
) )
173, 16sylan 459 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  =/=  (/) )  ->  -.  (/) 
e.  ( ( nei `  J ) `  S
) )
18173adant2 979 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  -.  (/)  e.  ( ( nei `  J
) `  S )
)
197tpnei 16690 . . . . . . 7  |-  ( J  e.  Top  ->  ( S  C_  U. J  <->  U. J  e.  ( ( nei `  J
) `  S )
) )
2019biimpa 472 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  U. J  e.  ( ( nei `  J
) `  S )
)
214, 6, 20syl2anc 645 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  U. J  e.  ( ( nei `  J
) `  S )
)
222, 21eqeltrd 2327 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  X  e.  ( ( nei `  J
) `  S )
)
23223adant3 980 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  X  e.  ( ( nei `  J
) `  S )
)
2415, 18, 233jca 1137 . 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 3538 . . . . 5  |-  ( x  e.  ~P X  ->  x  C_  X )
264ad2antrr 709 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  J  e.  Top )
27 simprl 735 . . . . . . . 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 736 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  y  C_  x )
29 simplr 734 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  x  C_  X
)
302ad2antrr 709 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  X  =  U. J )
3129, 30sseqtrd 3135 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  x  C_  U. J
)
327ssnei2 16685 . . . . . . . 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 1188 . . . . . . 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 601 . . . . . 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 2629 . . . . 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 462 . . . 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 2588 . . 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 980 . 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 16694 . . . . . 6  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S )  /\  y  e.  ( ( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
)
40393expib 1159 . . . . 5  |-  ( J  e.  Top  ->  (
( x  e.  ( ( nei `  J
) `  S )  /\  y  e.  (
( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
) )
413, 40syl 17 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( (
x  e.  ( ( nei `  J ) `
 S )  /\  y  e.  ( ( nei `  J ) `  S ) )  -> 
( x  i^i  y
)  e.  ( ( nei `  J ) `
 S ) ) )
42413ad2ant1 981 . . 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 2596 . 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 17383 . 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 1141 1  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( ( nei `  J ) `
 S )  e.  ( Fil `  X
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   A.wral 2509   E.wrex 2510    i^i cin 3077    C_ wss 3078   (/)c0 3362   ~Pcpw 3530   U.cuni 3727   ` cfv 4592   Topctop 16463  TopOnctopon 16464   neicnei 16666   Filcfil 17372
This theorem is referenced by:  trnei  17419  neiflim  17501  hausflim  17508  flimcf  17509  flimclslem  17511  cnpflf2  17527  cnpflf  17528  fclsfnflim  17554  conttnf2  24728
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-top 16468  df-topon 16471  df-nei 16667  df-fbas 17352  df-fil 17373
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