MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unitgOLD Structured version   Unicode version

Theorem unitgOLD 19596
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.) Obsolete version of unitg 19595 as of 30-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
unitgOLD  |-  ( B  e.  V  ->  U. ( topGen `
 B )  = 
U. B )

Proof of Theorem unitgOLD
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 4255 . . . 4  |-  ( x  e.  U. ( topGen `  B )  <->  E. y  e.  ( topGen `  B )
x  e.  y )
2 eltg 19585 . . . . . 6  |-  ( B  e.  V  ->  (
y  e.  ( topGen `  B )  <->  y  C_  U. ( B  i^i  ~P y ) ) )
3 inss1 3714 . . . . . . . . 9  |-  ( B  i^i  ~P y ) 
C_  B
43unissi 4274 . . . . . . . 8  |-  U. ( B  i^i  ~P y ) 
C_  U. B
5 sstr 3507 . . . . . . . 8  |-  ( ( y  C_  U. ( B  i^i  ~P y )  /\  U. ( B  i^i  ~P y ) 
C_  U. B )  -> 
y  C_  U. B )
64, 5mpan2 671 . . . . . . 7  |-  ( y 
C_  U. ( B  i^i  ~P y )  ->  y  C_ 
U. B )
76sseld 3498 . . . . . 6  |-  ( y 
C_  U. ( B  i^i  ~P y )  ->  (
x  e.  y  ->  x  e.  U. B ) )
82, 7syl6bi 228 . . . . 5  |-  ( B  e.  V  ->  (
y  e.  ( topGen `  B )  ->  (
x  e.  y  ->  x  e.  U. B ) ) )
98rexlimdv 2947 . . . 4  |-  ( B  e.  V  ->  ( E. y  e.  ( topGen `
 B ) x  e.  y  ->  x  e.  U. B ) )
101, 9syl5bi 217 . . 3  |-  ( B  e.  V  ->  (
x  e.  U. ( topGen `
 B )  ->  x  e.  U. B ) )
11 bastg 19594 . . . . 5  |-  ( B  e.  V  ->  B  C_  ( topGen `  B )
)
1211unissd 4275 . . . 4  |-  ( B  e.  V  ->  U. B  C_ 
U. ( topGen `  B
) )
1312sseld 3498 . . 3  |-  ( B  e.  V  ->  (
x  e.  U. B  ->  x  e.  U. ( topGen `
 B ) ) )
1410, 13impbid 191 . 2  |-  ( B  e.  V  ->  (
x  e.  U. ( topGen `
 B )  <->  x  e.  U. B ) )
1514eqrdv 2454 1  |-  ( B  e.  V  ->  U. ( topGen `
 B )  = 
U. B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   E.wrex 2808    i^i cin 3470    C_ wss 3471   ~Pcpw 4015   U.cuni 4251   ` cfv 5594   topGenctg 14855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-topgen 14861
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator