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Theorem neifg 29781
Description: The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 20073. (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 20073 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  { x  e.  J  |  S  C_  x }  e.  (
fBas `  X )
)
3 fgval 20101 . . 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 16 . 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 4008 . . . . . . 7  |-  ( t  =  u  ->  ~P t  =  ~P u
)
65ineq2d 3695 . . . . . 6  |-  ( t  =  u  ->  ( { x  e.  J  |  S  C_  x }  i^i  ~P t )  =  ( { x  e.  J  |  S  C_  x }  i^i  ~P u
) )
76neeq1d 2739 . . . . 5  |-  ( t  =  u  ->  (
( { x  e.  J  |  S  C_  x }  i^i  ~P t
)  =/=  (/)  <->  ( {
x  e.  J  |  S  C_  x }  i^i  ~P u )  =/=  (/) ) )
87elrab 3256 . . . 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 selpw 4012 . . . . . . 7  |-  ( u  e.  ~P X  <->  u  C_  X
)
109a1i 11 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  (
u  e.  ~P X  <->  u 
C_  X ) )
11 n0 3789 . . . . . . . 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
) )
12 elin 3682 . . . . . . . . . 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 ) )
13 sseq2 3521 . . . . . . . . . . . 12  |-  ( x  =  z  ->  ( S  C_  x  <->  S  C_  z
) )
1413elrab 3256 . . . . . . . . . . 11  |-  ( z  e.  { x  e.  J  |  S  C_  x }  <->  ( z  e.  J  /\  S  C_  z ) )
15 selpw 4012 . . . . . . . . . . 11  |-  ( z  e.  ~P u  <->  z  C_  u )
1614, 15anbi12i 697 . . . . . . . . . 10  |-  ( ( z  e.  { x  e.  J  |  S  C_  x }  /\  z  e.  ~P u )  <->  ( (
z  e.  J  /\  S  C_  z )  /\  z  C_  u ) )
1712, 16bitri 249 . . . . . . . . 9  |-  ( z  e.  ( { x  e.  J  |  S  C_  x }  i^i  ~P u )  <->  ( (
z  e.  J  /\  S  C_  z )  /\  z  C_  u ) )
1817exbii 1639 . . . . . . . 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 ) )
1911, 18bitri 249 . . . . . . 7  |-  ( ( { x  e.  J  |  S  C_  x }  i^i  ~P u )  =/=  (/) 
<->  E. z ( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u ) )
2019a1i 11 . . . . . 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 ) ) )
2110, 20anbi12d 710 . . . . 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 ) ) ) )
221isnei 19365 . . . . . . 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 ) ) ) )
23223adant3 1011 . . . . . 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 ) ) ) )
24 anass 649 . . . . . . . . 9  |-  ( ( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u
)  <->  ( z  e.  J  /\  ( S 
C_  z  /\  z  C_  u ) ) )
2524exbii 1639 . . . . . . . 8  |-  ( E. z ( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u )  <->  E. z
( z  e.  J  /\  ( S  C_  z  /\  z  C_  u ) ) )
26 df-rex 2815 . . . . . . . 8  |-  ( E. z  e.  J  ( S  C_  z  /\  z  C_  u )  <->  E. z
( z  e.  J  /\  ( S  C_  z  /\  z  C_  u ) ) )
2725, 26bitr4i 252 . . . . . . 7  |-  ( E. z ( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u )  <->  E. z  e.  J  ( S  C_  z  /\  z  C_  u ) )
2827anbi2i 694 . . . . . 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 ) ) )
2923, 28syl6rbbr 264 . . . . 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 ) ) )
3021, 29bitrd 253 . . . 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 )
) )
318, 30syl5bb 257 . . 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 )
) )
3231eqrdv 2459 . 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 )
)
334, 32eqtrd 2503 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( X filGen { x  e.  J  |  S  C_  x } )  =  ( ( nei `  J
) `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2657   E.wrex 2810   {crab 2813    i^i cin 3470    C_ wss 3471   (/)c0 3780   ~Pcpw 4005   U.cuni 4240   ` cfv 5581  (class class class)co 6277   fBascfbas 18172   filGencfg 18173   Topctop 19156   neicnei 19359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-fbas 18182  df-fg 18183  df-top 19161  df-nei 19360
This theorem is referenced by: (None)
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