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Theorem eltg 12148
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 12145 . . 3  |-  ( B  e.  V  ->  ( topGen `
 B )  =  { x  |  x 
C_  U. ( B  i^i  ~P x ) } )
21eleq2d 2187 . 2  |-  ( B  e.  V  ->  ( A  e.  ( topGen `  B )  <->  A  e.  { x  |  x  C_  U. ( B  i^i  ~P x ) } ) )
3 elex 2671 . . . 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 4034 . . . . . 6  |-  ( B  e.  V  ->  ( B  i^i  ~P A )  e.  _V )
6 uniexg 4331 . . . . . 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 4037 . . . . 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 3483 . . . . . . 7  |-  ( x  =  A  ->  ~P x  =  ~P A
)
1312ineq2d 3247 . . . . . 6  |-  ( x  =  A  ->  ( B  i^i  ~P x )  =  ( B  i^i  ~P A ) )
1413unieqd 3717 . . . . 5  |-  ( x  =  A  ->  U. ( B  i^i  ~P x )  =  U. ( B  i^i  ~P A ) )
1511, 14sseq12d 3098 . . . 4  |-  ( x  =  A  ->  (
x  C_  U. ( B  i^i  ~P x )  <-> 
A  C_  U. ( B  i^i  ~P A ) ) )
1615elabg 2803 . . 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 886 . 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 1316    e. wcel 1465   {cab 2103   _Vcvv 2660    i^i cin 3040    C_ wss 3041   ~Pcpw 3480   U.cuni 3706   ` cfv 5093   topGenctg 12062
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-topgen 12068
This theorem is referenced by:  eltg4i  12151  eltg3i  12152  bastg  12157  tgss  12159  eltop  12165
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