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

Theorem neifil 17571
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 16661 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
21adantr 451 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  X  =  U. J )
3 topontop 16660 . . . . . . . . 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 3215 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  S  C_ 
U. J )
7 eqid 2284 . . . . . . . . 9  |-  U. J  =  U. J
87neiuni 16855 . . . . . . . 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 2316 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  X  =  U. ( ( nei `  J ) `  S
) )
11 eqimss2 3232 . . . . . 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 3988 . . . . 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 16845 . . . . 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 16854 . . . . . . 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 2358 . . . 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 3634 . . . . 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 3215 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  x  C_  X
)  /\  ( y  e.  ( ( nei `  J
) `  S )  /\  y  C_  x ) )  ->  x  C_  U. J
)
327ssnei2 16849 . . . . . . . 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 2668 . . . . 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 2627 . . 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 16858 . . . . . 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 2635 . 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 17547 . 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 1623    e. wcel 1685    =/= wne 2447   A.wral 2544   E.wrex 2545    i^i cin 3152    C_ wss 3153   (/)c0 3456   ~Pcpw 3626   U.cuni 3828   ` cfv 5221   Topctop 16627  TopOnctopon 16628   neicnei 16830   Filcfil 17536
This theorem is referenced by:  trnei  17583  neiflim  17665  hausflim  17672  flimcf  17673  flimclslem  17675  cnpflf2  17691  cnpflf  17692  fclsfnflim  17718  conttnf2  24973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-top 16632  df-topon 16635  df-nei 16831  df-fbas 17516  df-fil 17537
  Copyright terms: Public domain W3C validator