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Theorem tgval 17012
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 2956 . 2  |-  ( B  e.  V  ->  B  e.  _V )
2 uniexg 4698 . . 3  |-  ( B  e.  V  ->  U. B  e.  _V )
3 abssexg 4376 . . 3  |-  ( U. B  e.  _V  ->  { x  |  ( x 
C_  U. B  /\  x  C_ 
U. ~P x ) }  e.  _V )
4 uniin 4027 . . . . . . 7  |-  U. ( B  i^i  ~P x ) 
C_  ( U. B  i^i  U. ~P x )
5 sstr 3348 . . . . . . 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 3555 . . . . . 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 3407 . . . 4  |-  { x  |  x  C_  U. ( B  i^i  ~P x ) }  C_  { x  |  ( x  C_  U. B  /\  x  C_  U. ~P x ) }
10 ssexg 4341 . . . 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 3527 . . . . . 6  |-  ( y  =  B  ->  (
y  i^i  ~P x
)  =  ( B  i^i  ~P x ) )
1413unieqd 4018 . . . . 5  |-  ( y  =  B  ->  U. (
y  i^i  ~P x
)  =  U. ( B  i^i  ~P x ) )
1514sseq2d 3368 . . . 4  |-  ( y  =  B  ->  (
x  C_  U. (
y  i^i  ~P x
)  <->  x  C_  U. ( B  i^i  ~P x ) ) )
1615abbidv 2549 . . 3  |-  ( y  =  B  ->  { x  |  x  C_  U. (
y  i^i  ~P x
) }  =  {
x  |  x  C_  U. ( B  i^i  ~P x ) } )
17 df-topgen 13659 . . 3  |-  topGen  =  ( y  e.  _V  |->  { x  |  x  C_  U. ( y  i^i  ~P x ) } )
1816, 17fvmptg 5796 . 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 1652    e. wcel 1725   {cab 2421   _Vcvv 2948    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   ` cfv 5446   topGenctg 13657
This theorem is referenced by:  tgval2  17013  eltg  17014  tgdif0  17049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-topgen 13659
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