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Theorem unitg 12245
Description: The topology generated by a basis  B is a topology on  U. B. Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class  TopBases completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.)
Assertion
Ref Expression
unitg  |-  ( B  e.  V  ->  U. ( topGen `
 B )  = 
U. B )

Proof of Theorem unitg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 tg1 12242 . . . . . 6  |-  ( x  e.  ( topGen `  B
)  ->  x  C_  U. B
)
2 velpw 3517 . . . . . 6  |-  ( x  e.  ~P U. B  <->  x 
C_  U. B )
31, 2sylibr 133 . . . . 5  |-  ( x  e.  ( topGen `  B
)  ->  x  e.  ~P U. B )
43ssriv 3101 . . . 4  |-  ( topGen `  B )  C_  ~P U. B
5 sspwuni 3897 . . . 4  |-  ( (
topGen `  B )  C_  ~P U. B  <->  U. ( topGen `
 B )  C_  U. B )
64, 5mpbi 144 . . 3  |-  U. ( topGen `
 B )  C_  U. B
76a1i 9 . 2  |-  ( B  e.  V  ->  U. ( topGen `
 B )  C_  U. B )
8 bastg 12244 . . 3  |-  ( B  e.  V  ->  B  C_  ( topGen `  B )
)
98unissd 3760 . 2  |-  ( B  e.  V  ->  U. B  C_ 
U. ( topGen `  B
) )
107, 9eqssd 3114 1  |-  ( B  e.  V  ->  U. ( topGen `
 B )  = 
U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480    C_ wss 3071   ~Pcpw 3510   U.cuni 3736   ` cfv 5123   topGenctg 12149
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-topgen 12155
This theorem is referenced by:  tgcl  12247  tgtopon  12249  txtopon  12445  uniretop  12708
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