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Theorem neifg 28517
Description: The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 19374. (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 19374 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  { x  e.  J  |  S  C_  x }  e.  (
fBas `  X )
)
3 fgval 19402 . . 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 3860 . . . . . . 7  |-  ( t  =  u  ->  ~P t  =  ~P u
)
65ineq2d 3549 . . . . . 6  |-  ( t  =  u  ->  ( { x  e.  J  |  S  C_  x }  i^i  ~P t )  =  ( { x  e.  J  |  S  C_  x }  i^i  ~P u
) )
76neeq1d 2619 . . . . 5  |-  ( t  =  u  ->  (
( { x  e.  J  |  S  C_  x }  i^i  ~P t
)  =/=  (/)  <->  ( {
x  e.  J  |  S  C_  x }  i^i  ~P u )  =/=  (/) ) )
87elrab 3114 . . . 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 3864 . . . . . . 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 3643 . . . . . . . 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 3536 . . . . . . . . . 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 3375 . . . . . . . . . . . 12  |-  ( x  =  z  ->  ( S  C_  x  <->  S  C_  z
) )
1413elrab 3114 . . . . . . . . . . 11  |-  ( z  e.  { x  e.  J  |  S  C_  x }  <->  ( z  e.  J  /\  S  C_  z ) )
15 selpw 3864 . . . . . . . . . . 11  |-  ( z  e.  ~P u  <->  z  C_  u )
1614, 15anbi12i 692 . . . . . . . . . 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 705 . . . . 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 18666 . . . . . . 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 1003 . . . . . 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 644 . . . . . . . . 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 2719 . . . . . . . 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 689 . . . . . 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 2439 . 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 2473 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 960    = wceq 1364   E.wex 1591    e. wcel 1761    =/= wne 2604   E.wrex 2714   {crab 2717    i^i cin 3324    C_ wss 3325   (/)c0 3634   ~Pcpw 3857   U.cuni 4088   ` cfv 5415  (class class class)co 6090   fBascfbas 17763   filGencfg 17764   Topctop 18457   neicnei 18660
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-fbas 17773  df-fg 17774  df-top 18462  df-nei 18661
This theorem is referenced by: (None)
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