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Theorem eltg 12231
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 12228 . . 3  |-  ( B  e.  V  ->  ( topGen `
 B )  =  { x  |  x 
C_  U. ( B  i^i  ~P x ) } )
21eleq2d 2209 . 2  |-  ( B  e.  V  ->  ( A  e.  ( topGen `  B )  <->  A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) } ) )
3 elex 2697 . . . 4  |-  ( A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) }  ->  A  e.  _V )
43adantl 275 . . 3  |-  ( ( B  e.  V  /\  A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) } )  ->  A  e.  _V )
5 inex1g 4064 . . . . . 6  |-  ( B  e.  V  ->  ( B  i^i  ~P A )  e.  _V )
6 uniexg 4361 . . . . . 6  |-  ( ( B  i^i  ~P A
)  e.  _V  ->  U. ( B  i^i  ~P A )  e.  _V )
75, 6syl 14 . . . . 5  |-  ( B  e.  V  ->  U. ( B  i^i  ~P A )  e.  _V )
8 ssexg 4067 . . . . 5  |-  ( ( A  C_  U. ( B  i^i  ~P A )  /\  U. ( B  i^i  ~P A )  e.  _V )  ->  A  e.  _V )
97, 8sylan2 284 . . . 4  |-  ( ( A  C_  U. ( B  i^i  ~P A )  /\  B  e.  V
)  ->  A  e.  _V )
109ancoms 266 . . 3  |-  ( ( B  e.  V  /\  A  C_  U. ( B  i^i  ~P A ) )  ->  A  e.  _V )
11 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
12 pweq 3513 . . . . . . 7  |-  ( x  =  A  ->  ~P x  =  ~P A
)
1312ineq2d 3277 . . . . . 6  |-  ( x  =  A  ->  ( B  i^i  ~P x )  =  ( B  i^i  ~P A ) )
1413unieqd 3747 . . . . 5  |-  ( x  =  A  ->  U. ( B  i^i  ~P x )  =  U. ( B  i^i  ~P A ) )
1511, 14sseq12d 3128 . . . 4  |-  ( x  =  A  ->  (
x  C_  U. ( B  i^i  ~P x )  <-> 
A  C_  U. ( B  i^i  ~P A ) ) )
1615elabg 2830 . . 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 187 1  |-  ( B  e.  V  ->  ( A  e.  ( topGen `  B )  <->  A  C_  U. ( B  i^i  ~P A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480   {cab 2125   _Vcvv 2686    i^i cin 3070    C_ wss 3071   ~Pcpw 3510   U.cuni 3736   ` cfv 5123   topGenctg 12145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-topgen 12151
This theorem is referenced by:  eltg4i  12234  eltg3i  12235  bastg  12240  tgss  12242  eltop  12248
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