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Theorem tgval 16709
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 2809 . 2  |-  ( B  e.  V  ->  B  e.  _V )
2 uniexg 4533 . . 3  |-  ( B  e.  V  ->  U. B  e.  _V )
3 abssexg 4211 . . 3  |-  ( U. B  e.  _V  ->  { x  |  ( x 
C_  U. B  /\  x  C_ 
U. ~P x ) }  e.  _V )
4 uniin 3863 . . . . . . 7  |-  U. ( B  i^i  ~P x ) 
C_  ( U. B  i^i  U. ~P x )
5 sstr 3200 . . . . . . 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 652 . . . . . 6  |-  ( x 
C_  U. ( B  i^i  ~P x )  ->  x  C_  ( U. B  i^i  U. ~P x ) )
7 ssin 3404 . . . . . 6  |-  ( ( x  C_  U. B  /\  x  C_  U. ~P x
)  <->  x  C_  ( U. B  i^i  U. ~P x
) )
86, 7sylibr 203 . . . . 5  |-  ( x 
C_  U. ( B  i^i  ~P x )  ->  (
x  C_  U. B  /\  x  C_  U. ~P x
) )
98ss2abi 3258 . . . 4  |-  { x  |  x  C_  U. ( B  i^i  ~P x ) }  C_  { x  |  ( x  C_  U. B  /\  x  C_  U. ~P x ) }
10 ssexg 4176 . . . 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 651 . . 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 18 . 2  |-  ( B  e.  V  ->  { x  |  x  C_  U. ( B  i^i  ~P x ) }  e.  _V )
13 ineq1 3376 . . . . . 6  |-  ( y  =  B  ->  (
y  i^i  ~P x
)  =  ( B  i^i  ~P x ) )
1413unieqd 3854 . . . . 5  |-  ( y  =  B  ->  U. (
y  i^i  ~P x
)  =  U. ( B  i^i  ~P x ) )
1514sseq2d 3219 . . . 4  |-  ( y  =  B  ->  (
x  C_  U. (
y  i^i  ~P x
)  <->  x  C_  U. ( B  i^i  ~P x ) ) )
1615abbidv 2410 . . 3  |-  ( y  =  B  ->  { x  |  x  C_  U. (
y  i^i  ~P x
) }  =  {
x  |  x  C_  U. ( B  i^i  ~P x ) } )
17 df-topgen 13360 . . 3  |-  topGen  =  ( y  e.  _V  |->  { x  |  x  C_  U. ( y  i^i  ~P x ) } )
1816, 17fvmptg 5616 . 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 642 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 358    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   ` cfv 5271   topGenctg 13358
This theorem is referenced by:  tgval2  16710  eltg  16711  tgdif0  16746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-topgen 13360
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