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Theorem tgval 16943
Description: The topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
Assertion
Ref Expression
tgval  |-  ( B  e.  V  ->  ( topGen `
 B )  =  { x  |  x 
C_  U. ( B  i^i  ~P x ) } )
Distinct variable groups:    x, B    x, V

Proof of Theorem tgval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elex 2907 . 2  |-  ( B  e.  V  ->  B  e.  _V )
2 uniexg 4646 . . 3  |-  ( B  e.  V  ->  U. B  e.  _V )
3 abssexg 4325 . . 3  |-  ( U. B  e.  _V  ->  { x  |  ( x 
C_  U. B  /\  x  C_ 
U. ~P x ) }  e.  _V )
4 uniin 3977 . . . . . . 7  |-  U. ( B  i^i  ~P x ) 
C_  ( U. B  i^i  U. ~P x )
5 sstr 3299 . . . . . . 7  |-  ( ( x  C_  U. ( B  i^i  ~P x )  /\  U. ( B  i^i  ~P x ) 
C_  ( U. B  i^i  U. ~P x ) )  ->  x  C_  ( U. B  i^i  U. ~P x ) )
64, 5mpan2 653 . . . . . 6  |-  ( x 
C_  U. ( B  i^i  ~P x )  ->  x  C_  ( U. B  i^i  U. ~P x ) )
7 ssin 3506 . . . . . 6  |-  ( ( x  C_  U. B  /\  x  C_  U. ~P x
)  <->  x  C_  ( U. B  i^i  U. ~P x
) )
86, 7sylibr 204 . . . . 5  |-  ( x 
C_  U. ( B  i^i  ~P x )  ->  (
x  C_  U. B  /\  x  C_  U. ~P x
) )
98ss2abi 3358 . . . 4  |-  { x  |  x  C_  U. ( B  i^i  ~P x ) }  C_  { x  |  ( x  C_  U. B  /\  x  C_  U. ~P x ) }
10 ssexg 4290 . . . 4  |-  ( ( { x  |  x 
C_  U. ( B  i^i  ~P x ) }  C_  { x  |  ( x 
C_  U. B  /\  x  C_ 
U. ~P x ) }  /\  { x  |  ( x  C_  U. B  /\  x  C_  U. ~P x ) }  e.  _V )  ->  { x  |  x  C_ 
U. ( B  i^i  ~P x ) }  e.  _V )
119, 10mpan 652 . . 3  |-  ( { x  |  ( x 
C_  U. B  /\  x  C_ 
U. ~P x ) }  e.  _V  ->  { x  |  x  C_  U. ( B  i^i  ~P x ) }  e.  _V )
122, 3, 113syl 19 . 2  |-  ( B  e.  V  ->  { x  |  x  C_  U. ( B  i^i  ~P x ) }  e.  _V )
13 ineq1 3478 . . . . . 6  |-  ( y  =  B  ->  (
y  i^i  ~P x
)  =  ( B  i^i  ~P x ) )
1413unieqd 3968 . . . . 5  |-  ( y  =  B  ->  U. (
y  i^i  ~P x
)  =  U. ( B  i^i  ~P x ) )
1514sseq2d 3319 . . . 4  |-  ( y  =  B  ->  (
x  C_  U. (
y  i^i  ~P x
)  <->  x  C_  U. ( B  i^i  ~P x ) ) )
1615abbidv 2501 . . 3  |-  ( y  =  B  ->  { x  |  x  C_  U. (
y  i^i  ~P x
) }  =  {
x  |  x  C_  U. ( B  i^i  ~P x ) } )
17 df-topgen 13594 . . 3  |-  topGen  =  ( y  e.  _V  |->  { x  |  x  C_  U. ( y  i^i  ~P x ) } )
1816, 17fvmptg 5743 . 2  |-  ( ( B  e.  _V  /\  { x  |  x  C_  U. ( B  i^i  ~P x ) }  e.  _V )  ->  ( topGen `  B )  =  {
x  |  x  C_  U. ( B  i^i  ~P x ) } )
191, 12, 18syl2anc 643 1  |-  ( B  e.  V  ->  ( topGen `
 B )  =  { x  |  x 
C_  U. ( B  i^i  ~P x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2373   _Vcvv 2899    i^i cin 3262    C_ wss 3263   ~Pcpw 3742   U.cuni 3957   ` cfv 5394   topGenctg 13592
This theorem is referenced by:  tgval2  16944  eltg  16945  tgdif0  16980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-topgen 13594
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