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Theorem unitg 19758
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 19755 . . . . . 6  |-  ( x  e.  ( topGen `  B
)  ->  x  C_  U. B
)
2 selpw 3961 . . . . . 6  |-  ( x  e.  ~P U. B  <->  x 
C_  U. B )
31, 2sylibr 212 . . . . 5  |-  ( x  e.  ( topGen `  B
)  ->  x  e.  ~P U. B )
43ssriv 3445 . . . 4  |-  ( topGen `  B )  C_  ~P U. B
5 sspwuni 4359 . . . 4  |-  ( (
topGen `  B )  C_  ~P U. B  <->  U. ( topGen `
 B )  C_  U. B )
64, 5mpbi 208 . . 3  |-  U. ( topGen `
 B )  C_  U. B
76a1i 11 . 2  |-  ( B  e.  V  ->  U. ( topGen `
 B )  C_  U. B )
8 bastg 19757 . . 3  |-  ( B  e.  V  ->  B  C_  ( topGen `  B )
)
98unissd 4214 . 2  |-  ( B  e.  V  ->  U. B  C_ 
U. ( topGen `  B
) )
107, 9eqssd 3458 1  |-  ( B  e.  V  ->  U. ( topGen `
 B )  = 
U. B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    C_ wss 3413   ~Pcpw 3954   U.cuni 4190   ` cfv 5568   topGenctg 15050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-topgen 15056
This theorem is referenced by:  tgcl  19761  tgtopon  19763  tgcmp  20192  2ndcsep  20250  txtopon  20382  ptuni  20385  xkouni  20390  prdstopn  20419  tgqtop  20503  alexsubb  20836  alexsubALTlem3  20839  alexsubALTlem4  20840  ptcmplem1  20842  uniretop  21559  fneval  30567  fnemeet1  30581  kelac2  35353
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