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Theorem eltg3 16716
Description: Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
eltg3  |-  ( B  e.  V  ->  ( A  e.  ( topGen `  B )  <->  E. x
( x  C_  B  /\  A  =  U. x ) ) )
Distinct variable groups:    x, A    x, B    x, V

Proof of Theorem eltg3
StepHypRef Expression
1 elfvdm 5570 . . . 4  |-  ( A  e.  ( topGen `  B
)  ->  B  e.  dom  topGen )
2 inex1g 4173 . . . 4  |-  ( B  e.  dom  topGen  ->  ( B  i^i  ~P A )  e.  _V )
31, 2syl 15 . . 3  |-  ( A  e.  ( topGen `  B
)  ->  ( B  i^i  ~P A )  e. 
_V )
4 eltg4i 16714 . . 3  |-  ( A  e.  ( topGen `  B
)  ->  A  =  U. ( B  i^i  ~P A ) )
5 inss1 3402 . . . . . . 7  |-  ( B  i^i  ~P A ) 
C_  B
6 sseq1 3212 . . . . . . 7  |-  ( x  =  ( B  i^i  ~P A )  ->  (
x  C_  B  <->  ( B  i^i  ~P A )  C_  B ) )
75, 6mpbiri 224 . . . . . 6  |-  ( x  =  ( B  i^i  ~P A )  ->  x  C_  B )
87biantrurd 494 . . . . 5  |-  ( x  =  ( B  i^i  ~P A )  ->  ( A  =  U. x  <->  ( x  C_  B  /\  A  =  U. x
) ) )
9 unieq 3852 . . . . . 6  |-  ( x  =  ( B  i^i  ~P A )  ->  U. x  =  U. ( B  i^i  ~P A ) )
109eqeq2d 2307 . . . . 5  |-  ( x  =  ( B  i^i  ~P A )  ->  ( A  =  U. x  <->  A  =  U. ( B  i^i  ~P A ) ) )
118, 10bitr3d 246 . . . 4  |-  ( x  =  ( B  i^i  ~P A )  ->  (
( x  C_  B  /\  A  =  U. x )  <->  A  =  U. ( B  i^i  ~P A ) ) )
1211spcegv 2882 . . 3  |-  ( ( B  i^i  ~P A
)  e.  _V  ->  ( A  =  U. ( B  i^i  ~P A )  ->  E. x ( x 
C_  B  /\  A  =  U. x ) ) )
133, 4, 12sylc 56 . 2  |-  ( A  e.  ( topGen `  B
)  ->  E. x
( x  C_  B  /\  A  =  U. x ) )
14 eltg3i 16715 . . . . 5  |-  ( ( B  e.  V  /\  x  C_  B )  ->  U. x  e.  ( topGen `
 B ) )
15 eleq1 2356 . . . . 5  |-  ( A  =  U. x  -> 
( A  e.  (
topGen `  B )  <->  U. x  e.  ( topGen `  B )
) )
1614, 15syl5ibrcom 213 . . . 4  |-  ( ( B  e.  V  /\  x  C_  B )  -> 
( A  =  U. x  ->  A  e.  (
topGen `  B ) ) )
1716expimpd 586 . . 3  |-  ( B  e.  V  ->  (
( x  C_  B  /\  A  =  U. x )  ->  A  e.  ( topGen `  B )
) )
1817exlimdv 1626 . 2  |-  ( B  e.  V  ->  ( E. x ( x  C_  B  /\  A  =  U. x )  ->  A  e.  ( topGen `  B )
) )
1913, 18impbid2 195 1  |-  ( B  e.  V  ->  ( A  e.  ( topGen `  B )  <->  E. x
( x  C_  B  /\  A  =  U. x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   dom cdm 4705   ` cfv 5271   topGenctg 13358
This theorem is referenced by:  tgval3  16717  tgtop  16727  eltop3  16730  tgidm  16734  bastop1  16747  tgrest  16906  tgcn  16998  txbasval  17317  opnmblALT  18974  mbfimaopnlem  19026  isfne3  26375  fneuni  26379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-topgen 13360
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