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Theorem tgval 17025
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 2966 . 2  |-  ( B  e.  V  ->  B  e.  _V )
2 uniexg 4709 . . 3  |-  ( B  e.  V  ->  U. B  e.  _V )
3 abssexg 4387 . . 3  |-  ( U. B  e.  _V  ->  { x  |  ( x 
C_  U. B  /\  x  C_ 
U. ~P x ) }  e.  _V )
4 uniin 4037 . . . . . . 7  |-  U. ( B  i^i  ~P x ) 
C_  ( U. B  i^i  U. ~P x )
5 sstr 3358 . . . . . . 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 654 . . . . . 6  |-  ( x 
C_  U. ( B  i^i  ~P x )  ->  x  C_  ( U. B  i^i  U. ~P x ) )
7 ssin 3565 . . . . . 6  |-  ( ( x  C_  U. B  /\  x  C_  U. ~P x
)  <->  x  C_  ( U. B  i^i  U. ~P x
) )
86, 7sylibr 205 . . . . 5  |-  ( x 
C_  U. ( B  i^i  ~P x )  ->  (
x  C_  U. B  /\  x  C_  U. ~P x
) )
98ss2abi 3417 . . . 4  |-  { x  |  x  C_  U. ( B  i^i  ~P x ) }  C_  { x  |  ( x  C_  U. B  /\  x  C_  U. ~P x ) }
10 ssexg 4352 . . . 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 653 . . 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 3537 . . . . . 6  |-  ( y  =  B  ->  (
y  i^i  ~P x
)  =  ( B  i^i  ~P x ) )
1413unieqd 4028 . . . . 5  |-  ( y  =  B  ->  U. (
y  i^i  ~P x
)  =  U. ( B  i^i  ~P x ) )
1514sseq2d 3378 . . . 4  |-  ( y  =  B  ->  (
x  C_  U. (
y  i^i  ~P x
)  <->  x  C_  U. ( B  i^i  ~P x ) ) )
1615abbidv 2552 . . 3  |-  ( y  =  B  ->  { x  |  x  C_  U. (
y  i^i  ~P x
) }  =  {
x  |  x  C_  U. ( B  i^i  ~P x ) } )
17 df-topgen 13672 . . 3  |-  topGen  =  ( y  e.  _V  |->  { x  |  x  C_  U. ( y  i^i  ~P x ) } )
1816, 17fvmptg 5807 . 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 644 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 360    = wceq 1653    e. wcel 1726   {cab 2424   _Vcvv 2958    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   ` cfv 5457   topGenctg 13670
This theorem is referenced by:  tgval2  17026  eltg  17027  tgdif0  17062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-topgen 13672
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