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Theorem eltg 19642
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 19640 . . 3  |-  ( B  e.  V  ->  ( topGen `
 B )  =  { x  |  x 
C_  U. ( B  i^i  ~P x ) } )
21eleq2d 2472 . 2  |-  ( B  e.  V  ->  ( A  e.  ( topGen `  B )  <->  A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) } ) )
3 elex 3067 . . . 4  |-  ( A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) }  ->  A  e.  _V )
43adantl 464 . . 3  |-  ( ( B  e.  V  /\  A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) } )  ->  A  e.  _V )
5 inex1g 4536 . . . . . 6  |-  ( B  e.  V  ->  ( B  i^i  ~P A )  e.  _V )
6 uniexg 6535 . . . . . 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 4539 . . . . 5  |-  ( ( A  C_  U. ( B  i^i  ~P A )  /\  U. ( B  i^i  ~P A )  e.  _V )  ->  A  e.  _V )
97, 8sylan2 472 . . . 4  |-  ( ( A  C_  U. ( B  i^i  ~P A )  /\  B  e.  V
)  ->  A  e.  _V )
109ancoms 451 . . 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 3957 . . . . . . 7  |-  ( x  =  A  ->  ~P x  =  ~P A
)
1312ineq2d 3640 . . . . . 6  |-  ( x  =  A  ->  ( B  i^i  ~P x )  =  ( B  i^i  ~P A ) )
1413unieqd 4200 . . . . 5  |-  ( x  =  A  ->  U. ( B  i^i  ~P x )  =  U. ( B  i^i  ~P A ) )
1511, 14sseq12d 3470 . . . 4  |-  ( x  =  A  ->  (
x  C_  U. ( B  i^i  ~P x )  <-> 
A  C_  U. ( B  i^i  ~P A ) ) )
1615elabg 3196 . . 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 901 . 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 1405    e. wcel 1842   {cab 2387   _Vcvv 3058    i^i cin 3412    C_ wss 3413   ~Pcpw 3954   U.cuni 4190   ` cfv 5525   topGenctg 14944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-iota 5489  df-fun 5527  df-fv 5533  df-topgen 14950
This theorem is referenced by:  eltg4i  19645  eltg3i  19646  bastg  19651  unitgOLD  19653  tgss  19654  eltop  19660  tgqtop  20397  isfne4  30556
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