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Theorem neifg 26423
Description: The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 17553. (Contributed by Jeff Hankins, 3-Sep-2009.)
Hypothesis
Ref Expression
neifg.1  |-  X  = 
U. J
Assertion
Ref Expression
neifg  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( X filGen { x  e.  J  |  S  C_  x } )  =  ( ( nei `  J
) `  S )
)
Distinct variable groups:    x, J    x, S    x, X

Proof of Theorem neifg
Dummy variables  u  t  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neifg.1 . . . 4  |-  X  = 
U. J
21opnfbas 17553 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  { x  e.  J  |  S  C_  x }  e.  (
fBas `  X )
)
3 fgval 17581 . . 3  |-  ( { x  e.  J  |  S  C_  x }  e.  ( fBas `  X )  ->  ( X filGen { x  e.  J  |  S  C_  x } )  =  { t  e.  ~P X  |  ( {
x  e.  J  |  S  C_  x }  i^i  ~P t )  =/=  (/) } )
42, 3syl 15 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( X filGen { x  e.  J  |  S  C_  x } )  =  {
t  e.  ~P X  |  ( { x  e.  J  |  S  C_  x }  i^i  ~P t )  =/=  (/) } )
5 pweq 3641 . . . . . . 7  |-  ( t  =  u  ->  ~P t  =  ~P u
)
65ineq2d 3383 . . . . . 6  |-  ( t  =  u  ->  ( { x  e.  J  |  S  C_  x }  i^i  ~P t )  =  ( { x  e.  J  |  S  C_  x }  i^i  ~P u
) )
76neeq1d 2472 . . . . 5  |-  ( t  =  u  ->  (
( { x  e.  J  |  S  C_  x }  i^i  ~P t
)  =/=  (/)  <->  ( {
x  e.  J  |  S  C_  x }  i^i  ~P u )  =/=  (/) ) )
87elrab 2936 . . . 4  |-  ( u  e.  { t  e. 
~P X  |  ( { x  e.  J  |  S  C_  x }  i^i  ~P t )  =/=  (/) }  <->  ( u  e. 
~P X  /\  ( { x  e.  J  |  S  C_  x }  i^i  ~P u )  =/=  (/) ) )
9 vex 2804 . . . . . . . 8  |-  u  e. 
_V
109elpw 3644 . . . . . . 7  |-  ( u  e.  ~P X  <->  u  C_  X
)
1110a1i 10 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  (
u  e.  ~P X  <->  u 
C_  X ) )
12 n0 3477 . . . . . . . 8  |-  ( ( { x  e.  J  |  S  C_  x }  i^i  ~P u )  =/=  (/) 
<->  E. z  z  e.  ( { x  e.  J  |  S  C_  x }  i^i  ~P u
) )
13 elin 3371 . . . . . . . . . 10  |-  ( z  e.  ( { x  e.  J  |  S  C_  x }  i^i  ~P u )  <->  ( z  e.  { x  e.  J  |  S  C_  x }  /\  z  e.  ~P u ) )
14 sseq2 3213 . . . . . . . . . . . 12  |-  ( x  =  z  ->  ( S  C_  x  <->  S  C_  z
) )
1514elrab 2936 . . . . . . . . . . 11  |-  ( z  e.  { x  e.  J  |  S  C_  x }  <->  ( z  e.  J  /\  S  C_  z ) )
16 vex 2804 . . . . . . . . . . . 12  |-  z  e. 
_V
1716elpw 3644 . . . . . . . . . . 11  |-  ( z  e.  ~P u  <->  z  C_  u )
1815, 17anbi12i 678 . . . . . . . . . 10  |-  ( ( z  e.  { x  e.  J  |  S  C_  x }  /\  z  e.  ~P u )  <->  ( (
z  e.  J  /\  S  C_  z )  /\  z  C_  u ) )
1913, 18bitri 240 . . . . . . . . 9  |-  ( z  e.  ( { x  e.  J  |  S  C_  x }  i^i  ~P u )  <->  ( (
z  e.  J  /\  S  C_  z )  /\  z  C_  u ) )
2019exbii 1572 . . . . . . . 8  |-  ( E. z  z  e.  ( { x  e.  J  |  S  C_  x }  i^i  ~P u )  <->  E. z
( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u ) )
2112, 20bitri 240 . . . . . . 7  |-  ( ( { x  e.  J  |  S  C_  x }  i^i  ~P u )  =/=  (/) 
<->  E. z ( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u ) )
2221a1i 10 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  (
( { x  e.  J  |  S  C_  x }  i^i  ~P u
)  =/=  (/)  <->  E. z
( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u ) ) )
2311, 22anbi12d 691 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  (
( u  e.  ~P X  /\  ( { x  e.  J  |  S  C_  x }  i^i  ~P u )  =/=  (/) )  <->  ( u  C_  X  /\  E. z
( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u ) ) ) )
241isnei 16856 . . . . . . 7  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( u  e.  ( ( nei `  J
) `  S )  <->  ( u  C_  X  /\  E. z  e.  J  ( S  C_  z  /\  z  C_  u ) ) ) )
25243adant3 975 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  (
u  e.  ( ( nei `  J ) `
 S )  <->  ( u  C_  X  /\  E. z  e.  J  ( S  C_  z  /\  z  C_  u ) ) ) )
26 anass 630 . . . . . . . . 9  |-  ( ( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u
)  <->  ( z  e.  J  /\  ( S 
C_  z  /\  z  C_  u ) ) )
2726exbii 1572 . . . . . . . 8  |-  ( E. z ( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u )  <->  E. z
( z  e.  J  /\  ( S  C_  z  /\  z  C_  u ) ) )
28 df-rex 2562 . . . . . . . 8  |-  ( E. z  e.  J  ( S  C_  z  /\  z  C_  u )  <->  E. z
( z  e.  J  /\  ( S  C_  z  /\  z  C_  u ) ) )
2927, 28bitr4i 243 . . . . . . 7  |-  ( E. z ( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u )  <->  E. z  e.  J  ( S  C_  z  /\  z  C_  u ) )
3029anbi2i 675 . . . . . 6  |-  ( ( u  C_  X  /\  E. z ( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u ) )  <->  ( u  C_  X  /\  E. z  e.  J  ( S  C_  z  /\  z  C_  u ) ) )
3125, 30syl6rbbr 255 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  (
( u  C_  X  /\  E. z ( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u ) )  <-> 
u  e.  ( ( nei `  J ) `
 S ) ) )
3223, 31bitrd 244 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  (
( u  e.  ~P X  /\  ( { x  e.  J  |  S  C_  x }  i^i  ~P u )  =/=  (/) )  <->  u  e.  ( ( nei `  J
) `  S )
) )
338, 32syl5bb 248 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  (
u  e.  { t  e.  ~P X  | 
( { x  e.  J  |  S  C_  x }  i^i  ~P t
)  =/=  (/) }  <->  u  e.  ( ( nei `  J
) `  S )
) )
3433eqrdv 2294 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  { t  e.  ~P X  | 
( { x  e.  J  |  S  C_  x }  i^i  ~P t
)  =/=  (/) }  =  ( ( nei `  J
) `  S )
)
354, 34eqtrd 2328 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( X filGen { x  e.  J  |  S  C_  x } )  =  ( ( nei `  J
) `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   U.cuni 3843   ` cfv 5271  (class class class)co 5874   Topctop 16647   neicnei 16850   fBascfbas 17534   filGencfg 17535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-top 16652  df-nei 16851  df-fbas 17536  df-fg 17537
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