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Theorem eltg 19984
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 19982 . . 3  |-  ( B  e.  V  ->  ( topGen `
 B )  =  { x  |  x 
C_  U. ( B  i^i  ~P x ) } )
21eleq2d 2516 . 2  |-  ( B  e.  V  ->  ( A  e.  ( topGen `  B )  <->  A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) } ) )
3 elex 3056 . . . 4  |-  ( A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) }  ->  A  e.  _V )
43adantl 468 . . 3  |-  ( ( B  e.  V  /\  A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) } )  ->  A  e.  _V )
5 inex1g 4549 . . . . . 6  |-  ( B  e.  V  ->  ( B  i^i  ~P A )  e.  _V )
6 uniexg 6593 . . . . . 6  |-  ( ( B  i^i  ~P A
)  e.  _V  ->  U. ( B  i^i  ~P A )  e.  _V )
75, 6syl 17 . . . . 5  |-  ( B  e.  V  ->  U. ( B  i^i  ~P A )  e.  _V )
8 ssexg 4552 . . . . 5  |-  ( ( A  C_  U. ( B  i^i  ~P A )  /\  U. ( B  i^i  ~P A )  e.  _V )  ->  A  e.  _V )
97, 8sylan2 477 . . . 4  |-  ( ( A  C_  U. ( B  i^i  ~P A )  /\  B  e.  V
)  ->  A  e.  _V )
109ancoms 455 . . 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 3956 . . . . . . 7  |-  ( x  =  A  ->  ~P x  =  ~P A
)
1312ineq2d 3636 . . . . . 6  |-  ( x  =  A  ->  ( B  i^i  ~P x )  =  ( B  i^i  ~P A ) )
1413unieqd 4211 . . . . 5  |-  ( x  =  A  ->  U. ( B  i^i  ~P x )  =  U. ( B  i^i  ~P A ) )
1511, 14sseq12d 3463 . . . 4  |-  ( x  =  A  ->  (
x  C_  U. ( B  i^i  ~P x )  <-> 
A  C_  U. ( B  i^i  ~P A ) ) )
1615elabg 3188 . . 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 912 . 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 257 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 188    = wceq 1446    e. wcel 1889   {cab 2439   _Vcvv 3047    i^i cin 3405    C_ wss 3406   ~Pcpw 3953   U.cuni 4201   ` cfv 5585   topGenctg 15348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5549  df-fun 5587  df-fv 5593  df-topgen 15354
This theorem is referenced by:  eltg4i  19987  eltg3i  19988  bastg  19993  unitgOLD  19995  tgss  19996  eltop  20002  tgqtop  20739  isfne4  31008
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