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Theorem eltg3 19548
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 5800 . . . 4  |-  ( A  e.  ( topGen `  B
)  ->  B  e.  dom  topGen )
2 inex1g 4508 . . . 4  |-  ( B  e.  dom  topGen  ->  ( B  i^i  ~P A )  e.  _V )
31, 2syl 16 . . 3  |-  ( A  e.  ( topGen `  B
)  ->  ( B  i^i  ~P A )  e. 
_V )
4 eltg4i 19546 . . 3  |-  ( A  e.  ( topGen `  B
)  ->  A  =  U. ( B  i^i  ~P A ) )
5 inss1 3632 . . . . . . 7  |-  ( B  i^i  ~P A ) 
C_  B
6 sseq1 3438 . . . . . . 7  |-  ( x  =  ( B  i^i  ~P A )  ->  (
x  C_  B  <->  ( B  i^i  ~P A )  C_  B ) )
75, 6mpbiri 233 . . . . . 6  |-  ( x  =  ( B  i^i  ~P A )  ->  x  C_  B )
87biantrurd 506 . . . . 5  |-  ( x  =  ( B  i^i  ~P A )  ->  ( A  =  U. x  <->  ( x  C_  B  /\  A  =  U. x
) ) )
9 unieq 4171 . . . . . 6  |-  ( x  =  ( B  i^i  ~P A )  ->  U. x  =  U. ( B  i^i  ~P A ) )
109eqeq2d 2396 . . . . 5  |-  ( x  =  ( B  i^i  ~P A )  ->  ( A  =  U. x  <->  A  =  U. ( B  i^i  ~P A ) ) )
118, 10bitr3d 255 . . . 4  |-  ( x  =  ( B  i^i  ~P A )  ->  (
( x  C_  B  /\  A  =  U. x )  <->  A  =  U. ( B  i^i  ~P A ) ) )
1211spcegv 3120 . . 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 60 . 2  |-  ( A  e.  ( topGen `  B
)  ->  E. x
( x  C_  B  /\  A  =  U. x ) )
14 eltg3i 19547 . . . . 5  |-  ( ( B  e.  V  /\  x  C_  B )  ->  U. x  e.  ( topGen `
 B ) )
15 eleq1 2454 . . . . 5  |-  ( A  =  U. x  -> 
( A  e.  (
topGen `  B )  <->  U. x  e.  ( topGen `  B )
) )
1614, 15syl5ibrcom 222 . . . 4  |-  ( ( B  e.  V  /\  x  C_  B )  -> 
( A  =  U. x  ->  A  e.  (
topGen `  B ) ) )
1716expimpd 601 . . 3  |-  ( B  e.  V  ->  (
( x  C_  B  /\  A  =  U. x )  ->  A  e.  ( topGen `  B )
) )
1817exlimdv 1732 . 2  |-  ( B  e.  V  ->  ( E. x ( x  C_  B  /\  A  =  U. x )  ->  A  e.  ( topGen `  B )
) )
1913, 18impbid2 204 1  |-  ( B  e.  V  ->  ( A  e.  ( topGen `  B )  <->  E. x
( x  C_  B  /\  A  =  U. x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399   E.wex 1620    e. wcel 1826   _Vcvv 3034    i^i cin 3388    C_ wss 3389   ~Pcpw 3927   U.cuni 4163   dom cdm 4913   ` cfv 5496   topGenctg 14845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-topgen 14851
This theorem is referenced by:  tgval3  19549  tgtop  19560  eltop3  19563  tgidm  19567  bastop1  19580  tgrest  19746  tgcn  19839  txbasval  20192  opnmblALT  22097  mbfimaopnlem  22147  isfne3  30327  fneuni  30331
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