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Theorem neifg 31020
Description: The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 20850. (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 20850 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  { x  e.  J  |  S  C_  x }  e.  (
fBas `  X )
)
3 fgval 20878 . . 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 17 . 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 3953 . . . . . . 7  |-  ( t  =  u  ->  ~P t  =  ~P u
)
65ineq2d 3633 . . . . . 6  |-  ( t  =  u  ->  ( { x  e.  J  |  S  C_  x }  i^i  ~P t )  =  ( { x  e.  J  |  S  C_  x }  i^i  ~P u
) )
76neeq1d 2682 . . . . 5  |-  ( t  =  u  ->  (
( { x  e.  J  |  S  C_  x }  i^i  ~P t
)  =/=  (/)  <->  ( {
x  e.  J  |  S  C_  x }  i^i  ~P u )  =/=  (/) ) )
87elrab 3195 . . . 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 3957 . . . . . . 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 3740 . . . . . . . 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 3616 . . . . . . . . . 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 3453 . . . . . . . . . . . 12  |-  ( x  =  z  ->  ( S  C_  x  <->  S  C_  z
) )
1413elrab 3195 . . . . . . . . . . 11  |-  ( z  e.  { x  e.  J  |  S  C_  x }  <->  ( z  e.  J  /\  S  C_  z ) )
15 selpw 3957 . . . . . . . . . . 11  |-  ( z  e.  ~P u  <->  z  C_  u )
1614, 15anbi12i 702 . . . . . . . . . 10  |-  ( ( z  e.  { x  e.  J  |  S  C_  x }  /\  z  e.  ~P u )  <->  ( (
z  e.  J  /\  S  C_  z )  /\  z  C_  u ) )
1712, 16bitri 253 . . . . . . . . 9  |-  ( z  e.  ( { x  e.  J  |  S  C_  x }  i^i  ~P u )  <->  ( (
z  e.  J  /\  S  C_  z )  /\  z  C_  u ) )
1817exbii 1717 . . . . . . . 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 253 . . . . . . 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 716 . . . . 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 20112 . . . . . . 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 1027 . . . . . 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 654 . . . . . . . . 9  |-  ( ( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u
)  <->  ( z  e.  J  /\  ( S 
C_  z  /\  z  C_  u ) ) )
2524exbii 1717 . . . . . . . 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 2742 . . . . . . . 8  |-  ( E. z  e.  J  ( S  C_  z  /\  z  C_  u )  <->  E. z
( z  e.  J  /\  ( S  C_  z  /\  z  C_  u ) ) )
2725, 26bitr4i 256 . . . . . . 7  |-  ( E. z ( ( z  e.  J  /\  S  C_  z )  /\  z  C_  u )  <->  E. z  e.  J  ( S  C_  z  /\  z  C_  u ) )
2827anbi2i 699 . . . . . 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 268 . . . . 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 257 . . . 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 261 . . 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 2448 . 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 2484 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 188    /\ wa 371    /\ w3a 984    = wceq 1443   E.wex 1662    e. wcel 1886    =/= wne 2621   E.wrex 2737   {crab 2740    i^i cin 3402    C_ wss 3403   (/)c0 3730   ~Pcpw 3950   U.cuni 4197   ` cfv 5581  (class class class)co 6288   fBascfbas 18951   filGencfg 18952   Topctop 19910   neicnei 20106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-fbas 18960  df-fg 18961  df-top 19914  df-nei 20107
This theorem is referenced by: (None)
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