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Theorem unitg 13229
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 13226 . . . . . 6  |-  ( x  e.  ( topGen `  B
)  ->  x  C_  U. B
)
2 velpw 3581 . . . . . 6  |-  ( x  e.  ~P U. B  <->  x 
C_  U. B )
31, 2sylibr 134 . . . . 5  |-  ( x  e.  ( topGen `  B
)  ->  x  e.  ~P U. B )
43ssriv 3159 . . . 4  |-  ( topGen `  B )  C_  ~P U. B
5 sspwuni 3968 . . . 4  |-  ( (
topGen `  B )  C_  ~P U. B  <->  U. ( topGen `
 B )  C_  U. B )
64, 5mpbi 145 . . 3  |-  U. ( topGen `
 B )  C_  U. B
76a1i 9 . 2  |-  ( B  e.  V  ->  U. ( topGen `
 B )  C_  U. B )
8 bastg 13228 . . 3  |-  ( B  e.  V  ->  B  C_  ( topGen `  B )
)
98unissd 3831 . 2  |-  ( B  e.  V  ->  U. B  C_ 
U. ( topGen `  B
) )
107, 9eqssd 3172 1  |-  ( B  e.  V  ->  U. ( topGen `
 B )  = 
U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148    C_ wss 3129   ~Pcpw 3574   U.cuni 3807   ` cfv 5212   topGenctg 12651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-topgen 12657
This theorem is referenced by:  tgcl  13231  tgtopon  13233  txtopon  13429  uniretop  13692
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