MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  neifil Unicode version

Theorem neifil 17865
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 16947 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
21adantr 452 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  X  =  U. J )
3 topontop 16946 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
43adantr 452 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  J  e.  Top )
5 simpr 448 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  S  C_  X )
65, 2sseqtrd 3344 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  S  C_ 
U. J )
7 eqid 2404 . . . . . . . . 9  |-  U. J  =  U. J
87neiuni 17141 . . . . . . . 8  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  U. J  =  U. ( ( nei `  J
) `  S )
)
94, 6, 8syl2anc 643 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  U. J  =  U. ( ( nei `  J ) `  S
) )
102, 9eqtrd 2436 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  X  =  U. ( ( nei `  J ) `  S
) )
11 eqimss2 3361 . . . . . 6  |-  ( X  =  U. ( ( nei `  J ) `
 S )  ->  U. ( ( nei `  J
) `  S )  C_  X )
1210, 11syl 16 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  U. (
( nei `  J
) `  S )  C_  X )
13 sspwuni 4136 . . . . 5  |-  ( ( ( nei `  J
) `  S )  C_ 
~P X  <->  U. (
( nei `  J
) `  S )  C_  X )
1412, 13sylibr 204 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  (
( nei `  J
) `  S )  C_ 
~P X )
15143adant3 977 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( ( nei `  J ) `
 S )  C_  ~P X )
16 0nnei 17131 . . . . 5  |-  ( ( J  e.  Top  /\  S  =/=  (/) )  ->  -.  (/) 
e.  ( ( nei `  J ) `  S
) )
173, 16sylan 458 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  =/=  (/) )  ->  -.  (/) 
e.  ( ( nei `  J ) `  S
) )
18173adant2 976 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  -.  (/)  e.  ( ( nei `  J
) `  S )
)
197tpnei 17140 . . . . . . 7  |-  ( J  e.  Top  ->  ( S  C_  U. J  <->  U. J  e.  ( ( nei `  J
) `  S )
) )
2019biimpa 471 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  U. J  e.  ( ( nei `  J
) `  S )
)
214, 6, 20syl2anc 643 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  U. J  e.  ( ( nei `  J
) `  S )
)
222, 21eqeltrd 2478 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  X  e.  ( ( nei `  J
) `  S )
)
23223adant3 977 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  X  e.  ( ( nei `  J
) `  S )
)
2415, 18, 233jca 1134 . 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 3767 . . . . 5  |-  ( x  e.  ~P X  ->  x  C_  X )
264ad2antrr 707 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  J  e.  Top )
27 simprl 733 . . . . . . 7  |-  ( ( ( ( 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 734 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  y  C_  x )
29 simplr 732 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  x  C_  X
)
302ad2antrr 707 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  X  =  U. J )
3129, 30sseqtrd 3344 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  x  C_  U. J
)
327ssnei2 17135 . . . . . . 7  |-  ( ( ( 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 1185 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  x  e.  ( ( nei `  J
) `  S )
)
3433rexlimdvaa 2791 . . . . 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
) ) )
3525, 34sylan2 461 . . . 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
) ) )
3635ralrimiva 2749 . . 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 )
) )
37363adant3 977 . 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 )
) )
38 innei 17144 . . . . . 6  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S )  /\  y  e.  ( ( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
)
39383expib 1156 . . . . 5  |-  ( J  e.  Top  ->  (
( x  e.  ( ( nei `  J
) `  S )  /\  y  e.  (
( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
) )
403, 39syl 16 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( (
x  e.  ( ( nei `  J ) `
 S )  /\  y  e.  ( ( nei `  J ) `  S ) )  -> 
( x  i^i  y
)  e.  ( ( nei `  J ) `
 S ) ) )
41403ad2ant1 978 . . 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 ) ) )
4241ralrimivv 2757 . 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 ) )
43 isfil2 17841 . 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 ) ) )
4424, 37, 42, 43syl3anbrc 1138 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   U.cuni 3975   ` cfv 5413   Topctop 16913  TopOnctopon 16914   neicnei 17116   Filcfil 17830
This theorem is referenced by:  trnei  17877  neiflim  17959  hausflim  17966  flimcf  17967  flimclslem  17969  cnpflf2  17985  cnpflf  17986  fclsfnflim  18012  neipcfilu  18279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-fbas 16654  df-top 16918  df-topon 16921  df-nei 17117  df-fil 17831
  Copyright terms: Public domain W3C validator