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Theorem brsiga 28511
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 df-brsiga 28510 . 2  |- 𝔅  =  (sigaGen `  ( topGen `
 ran  (,) )
)
2 retop 21452 . . 3  |-  ( topGen ` 
ran  (,) )  e.  Top
3 df-sigagen 28467 . . . . 5  |- sigaGen  =  ( x  e.  _V  |->  |^|
{ s  e.  (sigAlgebra ` 
U. x )  |  x  C_  s }
)
43funmpt2 5562 . . . 4  |-  Fun sigaGen
5 fvex 5815 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  _V
6 sigagensiga 28469 . . . . . 6  |-  ( (
topGen `  ran  (,) )  e.  _V  ->  (sigaGen `  ( topGen `
 ran  (,) )
)  e.  (sigAlgebra `  U. ( topGen `  ran  (,) )
) )
7 elrnsiga 28454 . . . . . 6  |-  ( (sigaGen `  ( topGen `  ran  (,) )
)  e.  (sigAlgebra `  U. ( topGen `  ran  (,) )
)  ->  (sigaGen `  ( topGen `
 ran  (,) )
)  e.  U. ran sigAlgebra )
85, 6, 7mp2b 10 . . . . 5  |-  (sigaGen `  ( topGen `
 ran  (,) )
)  e.  U. ran sigAlgebra
9 0elsiga 28442 . . . . 5  |-  ( (sigaGen `  ( topGen `  ran  (,) )
)  e.  U. ran sigAlgebra  ->  (/)  e.  (sigaGen `  ( topGen ` 
ran  (,) ) ) )
10 elfvdm 5831 . . . . 5  |-  ( (/)  e.  (sigaGen `  ( topGen ` 
ran  (,) ) )  -> 
( topGen `  ran  (,) )  e.  dom sigaGen )
118, 9, 10mp2b 10 . . . 4  |-  ( topGen ` 
ran  (,) )  e.  dom sigaGen
12 funfvima 6084 . . . 4  |-  ( ( Fun sigaGen  /\  ( topGen `  ran  (,) )  e.  dom sigaGen )  -> 
( ( topGen `  ran  (,) )  e.  Top  ->  (sigaGen `  ( topGen `  ran  (,) )
)  e.  (sigaGen " Top ) ) )
134, 11, 12mp2an 670 . . 3  |-  ( (
topGen `  ran  (,) )  e.  Top  ->  (sigaGen `  ( topGen `
 ran  (,) )
)  e.  (sigaGen " Top ) )
142, 13ax-mp 5 . 2  |-  (sigaGen `  ( topGen `
 ran  (,) )
)  e.  (sigaGen " Top )
151, 14eqeltri 2486 1  |- 𝔅  e.  (sigaGen " Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842   {crab 2757   _Vcvv 3058    C_ wss 3413   (/)c0 3737   U.cuni 4190   |^|cint 4226   dom cdm 4942   ran crn 4943   "cima 4945   Fun wfun 5519   ` cfv 5525   (,)cioo 11500   topGenctg 14944   Topctop 19578  sigAlgebracsiga 28435  sigaGencsigagen 28466  𝔅cbrsiga 28509
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 6530  ax-cnex 9498  ax-resscn 9499  ax-pre-lttri 9516  ax-pre-lttrn 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  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-nel 2601  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  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-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-ioo 11504  df-topgen 14950  df-top 19583  df-bases 19585  df-siga 28436  df-sigagen 28467  df-brsiga 28510
This theorem is referenced by: (None)
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