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Theorem eltg 19225
Description: Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
Assertion
Ref Expression
eltg  |-  ( B  e.  V  ->  ( A  e.  ( topGen `  B )  <->  A  C_  U. ( B  i^i  ~P A ) ) )

Proof of Theorem eltg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 tgval 19223 . . 3  |-  ( B  e.  V  ->  ( topGen `
 B )  =  { x  |  x 
C_  U. ( B  i^i  ~P x ) } )
21eleq2d 2537 . 2  |-  ( B  e.  V  ->  ( A  e.  ( topGen `  B )  <->  A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) } ) )
3 elex 3122 . . . 4  |-  ( A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) }  ->  A  e.  _V )
43adantl 466 . . 3  |-  ( ( B  e.  V  /\  A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) } )  ->  A  e.  _V )
5 inex1g 4590 . . . . . 6  |-  ( B  e.  V  ->  ( B  i^i  ~P A )  e.  _V )
6 uniexg 6579 . . . . . 6  |-  ( ( B  i^i  ~P A
)  e.  _V  ->  U. ( B  i^i  ~P A )  e.  _V )
75, 6syl 16 . . . . 5  |-  ( B  e.  V  ->  U. ( B  i^i  ~P A )  e.  _V )
8 ssexg 4593 . . . . 5  |-  ( ( A  C_  U. ( B  i^i  ~P A )  /\  U. ( B  i^i  ~P A )  e.  _V )  ->  A  e.  _V )
97, 8sylan2 474 . . . 4  |-  ( ( A  C_  U. ( B  i^i  ~P A )  /\  B  e.  V
)  ->  A  e.  _V )
109ancoms 453 . . 3  |-  ( ( B  e.  V  /\  A  C_  U. ( B  i^i  ~P A ) )  ->  A  e.  _V )
11 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
12 pweq 4013 . . . . . . 7  |-  ( x  =  A  ->  ~P x  =  ~P A
)
1312ineq2d 3700 . . . . . 6  |-  ( x  =  A  ->  ( B  i^i  ~P x )  =  ( B  i^i  ~P A ) )
1413unieqd 4255 . . . . 5  |-  ( x  =  A  ->  U. ( B  i^i  ~P x )  =  U. ( B  i^i  ~P A ) )
1511, 14sseq12d 3533 . . . 4  |-  ( x  =  A  ->  (
x  C_  U. ( B  i^i  ~P x )  <-> 
A  C_  U. ( B  i^i  ~P A ) ) )
1615elabg 3251 . . 3  |-  ( A  e.  _V  ->  ( A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) }  <->  A  C_  U. ( B  i^i  ~P A ) ) )
174, 10, 16pm5.21nd 898 . 2  |-  ( B  e.  V  ->  ( A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) }  <->  A  C_  U. ( B  i^i  ~P A ) ) )
182, 17bitrd 253 1  |-  ( B  e.  V  ->  ( A  e.  ( topGen `  B )  <->  A  C_  U. ( B  i^i  ~P A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   {cab 2452   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   ` cfv 5586   topGenctg 14689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-topgen 14695
This theorem is referenced by:  eltg4i  19228  eltg3i  19229  bastg  19234  unitg  19235  tgss  19236  eltop  19242  tgqtop  19948  isfne4  29741
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