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Theorem tg1 19635
Description: Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
Assertion
Ref Expression
tg1  |-  ( A  e.  ( topGen `  B
)  ->  A  C_  U. B
)

Proof of Theorem tg1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 5874 . 2  |-  ( A  e.  ( topGen `  B
)  ->  B  e.  dom  topGen )
2 eltg2 19629 . . 3  |-  ( B  e.  dom  topGen  ->  ( A  e.  ( topGen `  B )  <->  ( A  C_ 
U. B  /\  A. x  e.  A  E. y  e.  B  (
x  e.  y  /\  y  C_  A ) ) ) )
32simprbda 621 . 2  |-  ( ( B  e.  dom  topGen  /\  A  e.  ( topGen `  B )
)  ->  A  C_  U. B
)
41, 3mpancom 667 1  |-  ( A  e.  ( topGen `  B
)  ->  A  C_  U. B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   A.wral 2804   E.wrex 2805    C_ wss 3461   U.cuni 4235   dom cdm 4988   ` cfv 5570   topGenctg 14930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-topgen 14936
This theorem is referenced by:  unitg  19638  tgcl  19641  ontgval  30127
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