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Theorem brsiga 26765
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 26764 . 2  |- 𝔅  =  (sigaGen `  ( topGen `
 ran  (,) )
)
2 retop 20482 . . 3  |-  ( topGen ` 
ran  (,) )  e.  Top
3 df-sigagen 26750 . . . . 5  |- sigaGen  =  ( x  e.  _V  |->  |^|
{ s  e.  (sigAlgebra ` 
U. x )  |  x  C_  s }
)
43funmpt2 5566 . . . 4  |-  Fun sigaGen
5 fvex 5812 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  _V
6 sigagensiga 26752 . . . . . 6  |-  ( (
topGen `  ran  (,) )  e.  _V  ->  (sigaGen `  ( topGen `
 ran  (,) )
)  e.  (sigAlgebra `  U. ( topGen `  ran  (,) )
) )
7 elrnsiga 26737 . . . . . 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 26725 . . . . 5  |-  ( (sigaGen `  ( topGen `  ran  (,) )
)  e.  U. ran sigAlgebra  ->  (/)  e.  (sigaGen `  ( topGen ` 
ran  (,) ) ) )
10 elfvdm 5828 . . . . 5  |-  ( (/)  e.  (sigaGen `  ( topGen ` 
ran  (,) ) )  -> 
( topGen `  ran  (,) )  e.  dom sigaGen )
118, 9, 10mp2b 10 . . . 4  |-  ( topGen ` 
ran  (,) )  e.  dom sigaGen
12 funfvima 6064 . . . 4  |-  ( ( Fun sigaGen  /\  ( topGen `  ran  (,) )  e.  dom sigaGen )  -> 
( ( topGen `  ran  (,) )  e.  Top  ->  (sigaGen `  ( topGen `  ran  (,) )
)  e.  (sigaGen " Top ) ) )
134, 11, 12mp2an 672 . . 3  |-  ( (
topGen `  ran  (,) )  e.  Top  ->  (sigaGen `  ( topGen `
 ran  (,) )
)  e.  (sigaGen " Top ) )
142, 13ax-mp 5 . 2  |-  (sigaGen `  ( topGen `
 ran  (,) )
)  e.  (sigaGen " Top )
151, 14eqeltri 2538 1  |- 𝔅  e.  (sigaGen " Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   {crab 2803   _Vcvv 3078    C_ wss 3439   (/)c0 3748   U.cuni 4202   |^|cint 4239   dom cdm 4951   ran crn 4952   "cima 4954   Fun wfun 5523   ` cfv 5529   (,)cioo 11415   topGenctg 14499   Topctop 18640  sigAlgebracsiga 26718  sigaGencsigagen 26749  𝔅cbrsiga 26763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-pre-lttri 9471  ax-pre-lttrn 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-ioo 11419  df-topgen 14505  df-top 18645  df-bases 18647  df-siga 26719  df-sigagen 26750  df-brsiga 26764
This theorem is referenced by: (None)
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