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Theorem tgval 19901
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 3096 . 2  |-  ( B  e.  V  ->  B  e.  _V )
2 uniexg 6602 . . 3  |-  ( B  e.  V  ->  U. B  e.  _V )
3 abssexg 4610 . . 3  |-  ( U. B  e.  _V  ->  { x  |  ( x 
C_  U. B  /\  x  C_ 
U. ~P x ) }  e.  _V )
4 uniin 4242 . . . . . . 7  |-  U. ( B  i^i  ~P x ) 
C_  ( U. B  i^i  U. ~P x )
5 sstr 3478 . . . . . . 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 675 . . . . . 6  |-  ( x 
C_  U. ( B  i^i  ~P x )  ->  x  C_  ( U. B  i^i  U. ~P x ) )
7 ssin 3690 . . . . . 6  |-  ( ( x  C_  U. B  /\  x  C_  U. ~P x
)  <->  x  C_  ( U. B  i^i  U. ~P x
) )
86, 7sylibr 215 . . . . 5  |-  ( x 
C_  U. ( B  i^i  ~P x )  ->  (
x  C_  U. B  /\  x  C_  U. ~P x
) )
98ss2abi 3539 . . . 4  |-  { x  |  x  C_  U. ( B  i^i  ~P x ) }  C_  { x  |  ( x  C_  U. B  /\  x  C_  U. ~P x ) }
10 ssexg 4571 . . . 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 674 . . 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 3663 . . . . . 6  |-  ( y  =  B  ->  (
y  i^i  ~P x
)  =  ( B  i^i  ~P x ) )
1413unieqd 4232 . . . . 5  |-  ( y  =  B  ->  U. (
y  i^i  ~P x
)  =  U. ( B  i^i  ~P x ) )
1514sseq2d 3498 . . . 4  |-  ( y  =  B  ->  (
x  C_  U. (
y  i^i  ~P x
)  <->  x  C_  U. ( B  i^i  ~P x ) ) )
1615abbidv 2565 . . 3  |-  ( y  =  B  ->  { x  |  x  C_  U. (
y  i^i  ~P x
) }  =  {
x  |  x  C_  U. ( B  i^i  ~P x ) } )
17 df-topgen 15301 . . 3  |-  topGen  =  ( y  e.  _V  |->  { x  |  x  C_  U. ( y  i^i  ~P x ) } )
1816, 17fvmptg 5962 . 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 665 1  |-  ( B  e.  V  ->  ( topGen `
 B )  =  { x  |  x 
C_  U. ( B  i^i  ~P x ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   {cab 2414   _Vcvv 3087    i^i cin 3441    C_ wss 3442   ~Pcpw 3985   U.cuni 4222   ` cfv 5601   topGenctg 15295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-topgen 15301
This theorem is referenced by:  tgval2  19902  eltg  19903  tgdif0  19939
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