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Theorem br2base 28930
Description: The base set for the generator of the Borel sigma algebra on  ( RR  X.  RR ) is indeed  ( RR  X.  RR ). (Contributed by Thierry Arnoux, 22-Sep-2017.)
Assertion
Ref Expression
br2base  |-  U. ran  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  =  ( RR  X.  RR )
Distinct variable group:    x, y

Proof of Theorem br2base
StepHypRef Expression
1 brsigasspwrn 28846 . . . . . . . 8  |- 𝔅 
C_  ~P RR
21sseli 3466 . . . . . . 7  |-  ( x  e. 𝔅  ->  x  e.  ~P RR )
32elpwid 3995 . . . . . 6  |-  ( x  e. 𝔅  ->  x  C_  RR )
41sseli 3466 . . . . . . 7  |-  ( y  e. 𝔅  ->  y  e.  ~P RR )
54elpwid 3995 . . . . . 6  |-  ( y  e. 𝔅  ->  y  C_  RR )
6 xpss12 4960 . . . . . 6  |-  ( ( x  C_  RR  /\  y  C_  RR )  ->  (
x  X.  y ) 
C_  ( RR  X.  RR ) )
73, 5, 6syl2an 479 . . . . 5  |-  ( ( x  e. 𝔅  /\  y  e. 𝔅 )  ->  ( x  X.  y )  C_  ( RR  X.  RR ) )
8 vex 3090 . . . . . . 7  |-  x  e. 
_V
9 vex 3090 . . . . . . 7  |-  y  e. 
_V
108, 9xpex 6609 . . . . . 6  |-  ( x  X.  y )  e. 
_V
1110elpw 3991 . . . . 5  |-  ( ( x  X.  y )  e.  ~P ( RR 
X.  RR )  <->  ( x  X.  y )  C_  ( RR  X.  RR ) )
127, 11sylibr 215 . . . 4  |-  ( ( x  e. 𝔅  /\  y  e. 𝔅 )  ->  ( x  X.  y )  e.  ~P ( RR  X.  RR ) )
1312rgen2a 2859 . . 3  |-  A. x  e. 𝔅  A. y  e. 𝔅  ( x  X.  y
)  e.  ~P ( RR  X.  RR )
14 eqid 2429 . . . 4  |-  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  =  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )
1514rnmpt2ss 28116 . . 3  |-  ( A. x  e. 𝔅  A. y  e. 𝔅  ( x  X.  y
)  e.  ~P ( RR  X.  RR )  ->  ran  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  C_  ~P ( RR  X.  RR ) )
1613, 15ax-mp 5 . 2  |-  ran  (
x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  C_  ~P ( RR  X.  RR )
17 unibrsiga 28847 . . . . . 6  |-  U.𝔅  =  RR
18 brsigarn 28845 . . . . . . 7  |- 𝔅  e.  (sigAlgebra `  RR )
19 elrnsiga 28787 . . . . . . 7  |-  (𝔅  e.  (sigAlgebra `  RR )  -> 𝔅  e.  U. ran sigAlgebra )
20 unielsiga 28789 . . . . . . 7  |-  (𝔅  e.  U. ran sigAlgebra  ->  U.𝔅  e. 𝔅 )
2118, 19, 20mp2b 10 . . . . . 6  |-  U.𝔅  e. 𝔅
2217, 21eqeltrri 2514 . . . . 5  |-  RR  e. 𝔅
23 eqid 2429 . . . . 5  |-  ( RR 
X.  RR )  =  ( RR  X.  RR )
24 xpeq1 4868 . . . . . . 7  |-  ( x  =  RR  ->  (
x  X.  y )  =  ( RR  X.  y ) )
2524eqeq2d 2443 . . . . . 6  |-  ( x  =  RR  ->  (
( RR  X.  RR )  =  ( x  X.  y )  <->  ( RR  X.  RR )  =  ( RR  X.  y ) ) )
26 xpeq2 4869 . . . . . . 7  |-  ( y  =  RR  ->  ( RR  X.  y )  =  ( RR  X.  RR ) )
2726eqeq2d 2443 . . . . . 6  |-  ( y  =  RR  ->  (
( RR  X.  RR )  =  ( RR  X.  y )  <->  ( RR  X.  RR )  =  ( RR  X.  RR ) ) )
2825, 27rspc2ev 3199 . . . . 5  |-  ( ( RR  e. 𝔅  /\  RR  e. 𝔅  /\  ( RR  X.  RR )  =  ( RR  X.  RR ) )  ->  E. x  e. 𝔅  E. y  e. 𝔅  ( RR  X.  RR )  =  ( x  X.  y ) )
2922, 22, 23, 28mp3an 1360 . . . 4  |-  E. x  e. 𝔅  E. y  e. 𝔅  ( RR  X.  RR )  =  ( x  X.  y )
3014, 10elrnmpt2 6423 . . . 4  |-  ( ( RR  X.  RR )  e.  ran  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  <->  E. x  e. 𝔅  E. y  e. 𝔅  ( RR  X.  RR )  =  ( x  X.  y ) )
3129, 30mpbir 212 . . 3  |-  ( RR 
X.  RR )  e. 
ran  ( x  e. 𝔅 , 
y  e. 𝔅 
|->  ( x  X.  y
) )
32 elpwuni 4393 . . 3  |-  ( ( RR  X.  RR )  e.  ran  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  ->  ( ran  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  C_  ~P ( RR  X.  RR ) 
<-> 
U. ran  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  =  ( RR  X.  RR ) ) )
3331, 32ax-mp 5 . 2  |-  ( ran  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  C_  ~P ( RR  X.  RR ) 
<-> 
U. ran  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  =  ( RR  X.  RR ) )
3416, 33mpbi 211 1  |-  U. ran  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  =  ( RR  X.  RR )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783    C_ wss 3442   ~Pcpw 3985   U.cuni 4222    X. cxp 4852   ran crn 4855   ` cfv 5601    |-> cmpt2 6307   RRcr 9537  sigAlgebracsiga 28768  𝔅cbrsiga 28842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-pre-lttri 9612  ax-pre-lttrn 9613
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-ioo 11639  df-topgen 15301  df-top 19852  df-bases 19853  df-siga 28769  df-sigagen 28800  df-brsiga 28843
This theorem is referenced by:  sxbrsigalem5  28949  sxbrsiga  28951
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