Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brsiga Unicode version

Theorem brsiga 23516
Description: The Borel Algebra on real numbers is a Borel sigma algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
Assertion
Ref Expression
brsiga  |- 𝔅  e.  (sigaGen " Top )

Proof of Theorem brsiga
Dummy variables  x  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 retop 18272 . . 3  |-  ( topGen ` 
ran  (,) )  e.  Top
2 funmpt 5292 . . . . 5  |-  Fun  (
x  e.  _V  |->  |^|
{ s  e.  (sigAlgebra ` 
U. x )  |  x  C_  s }
)
3 df-sigagen 23502 . . . . . 6  |- sigaGen  =  ( x  e.  _V  |->  |^|
{ s  e.  (sigAlgebra ` 
U. x )  |  x  C_  s }
)
43funeqi 5277 . . . . 5  |-  ( Fun sigaGen  <->  Fun  ( x  e.  _V  |->  |^|
{ s  e.  (sigAlgebra ` 
U. x )  |  x  C_  s }
) )
52, 4mpbir 200 . . . 4  |-  Fun sigaGen
61elexi 2799 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  _V
7 sigagensiga 23504 . . . . . 6  |-  ( (
topGen `  ran  (,) )  e.  _V  ->  (sigaGen `  ( topGen `
 ran  (,) )
)  e.  (sigAlgebra `  U. ( topGen `  ran  (,) )
) )
8 issgon 23486 . . . . . . 7  |-  ( (sigaGen `  ( topGen `  ran  (,) )
)  e.  (sigAlgebra `  U. ( topGen `  ran  (,) )
)  <->  ( (sigaGen `  ( topGen `
 ran  (,) )
)  e.  U. ran sigAlgebra  /\  U. ( topGen `  ran  (,) )  =  U. (sigaGen `  ( topGen `
 ran  (,) )
) ) )
98simplbi 446 . . . . . 6  |-  ( (sigaGen `  ( topGen `  ran  (,) )
)  e.  (sigAlgebra `  U. ( topGen `  ran  (,) )
)  ->  (sigaGen `  ( topGen `
 ran  (,) )
)  e.  U. ran sigAlgebra )
106, 7, 9mp2b 9 . . . . 5  |-  (sigaGen `  ( topGen `
 ran  (,) )
)  e.  U. ran sigAlgebra
11 0elsiga 23477 . . . . 5  |-  ( (sigaGen `  ( topGen `  ran  (,) )
)  e.  U. ran sigAlgebra  ->  (/)  e.  (sigaGen `  ( topGen ` 
ran  (,) ) ) )
12 elfvdm 5556 . . . . 5  |-  ( (/)  e.  (sigaGen `  ( topGen ` 
ran  (,) ) )  -> 
( topGen `  ran  (,) )  e.  dom sigaGen )
1310, 11, 12mp2b 9 . . . 4  |-  ( topGen ` 
ran  (,) )  e.  dom sigaGen
14 funfvima 5755 . . . 4  |-  ( ( Fun sigaGen  /\  ( topGen `  ran  (,) )  e.  dom sigaGen )  -> 
( ( topGen `  ran  (,) )  e.  Top  ->  (sigaGen `  ( topGen `  ran  (,) )
)  e.  (sigaGen " Top ) ) )
155, 13, 14mp2an 653 . . 3  |-  ( (
topGen `  ran  (,) )  e.  Top  ->  (sigaGen `  ( topGen `
 ran  (,) )
)  e.  (sigaGen " Top ) )
161, 15ax-mp 8 . 2  |-  (sigaGen `  ( topGen `
 ran  (,) )
)  e.  (sigaGen " Top )
17 df-brsiga 23515 . . 3  |- 𝔅  =  (sigaGen `  ( topGen `
 ran  (,) )
)
1817eleq1i 2348 . 2  |-  (𝔅  e.  (sigaGen " Top )  <->  (sigaGen `  ( topGen `
 ran  (,) )
)  e.  (sigaGen " Top ) )
1916, 18mpbir 200 1  |- 𝔅  e.  (sigaGen " Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1625    e. wcel 1686   {crab 2549   _Vcvv 2790    C_ wss 3154   (/)c0 3457   U.cuni 3829   |^|cint 3864    e. cmpt 4079   dom cdm 4691   ran crn 4692   "cima 4694   Fun wfun 5251   ` cfv 5257   (,)cioo 10658   topGenctg 13344   Topctop 16633  sigAlgebracsiga 23470  sigaGencsigagen 23501  𝔅cbrsiga 23514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-pre-lttri 8813  ax-pre-lttrn 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-ioo 10662  df-topgen 13346  df-top 16638  df-bases 16640  df-siga 23471  df-sigagen 23502  df-brsiga 23515
  Copyright terms: Public domain W3C validator