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Theorem eltg3 16700
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 5554 . . . 4  |-  ( A  e.  ( topGen `  B
)  ->  B  e.  dom  topGen )
2 inex1g 4157 . . . 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 16698 . . 3  |-  ( A  e.  ( topGen `  B
)  ->  A  =  U. ( B  i^i  ~P A ) )
5 inss1 3389 . . . . . . 7  |-  ( B  i^i  ~P A ) 
C_  B
6 sseq1 3199 . . . . . . 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 3836 . . . . . 6  |-  ( x  =  ( B  i^i  ~P A )  ->  U. x  =  U. ( B  i^i  ~P A ) )
109eqeq2d 2294 . . . . 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 2869 . . 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 16699 . . . . 5  |-  ( ( B  e.  V  /\  x  C_  B )  ->  U. x  e.  ( topGen `
 B ) )
15 eleq1 2343 . . . . 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 1664 . 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 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   dom cdm 4689   ` cfv 5255   topGenctg 13342
This theorem is referenced by:  tgval3  16701  tgtop  16711  eltop3  16714  tgidm  16718  bastop1  16731  tgrest  16890  tgcn  16982  txbasval  17301  opnmblALT  18958  mbfimaopnlem  19010  isfne3  26272  fneuni  26276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-topgen 13344
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