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Theorem br2base 24415
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 24337 . . . . . . . 8  |- 𝔅 
C_  ~P RR
21sseli 3289 . . . . . . 7  |-  ( x  e. 𝔅  ->  x  e.  ~P RR )
32elpwid 3753 . . . . . 6  |-  ( x  e. 𝔅  ->  x  C_  RR )
41sseli 3289 . . . . . . 7  |-  ( y  e. 𝔅  ->  y  e.  ~P RR )
54elpwid 3753 . . . . . 6  |-  ( y  e. 𝔅  ->  y  C_  RR )
6 xpss12 4923 . . . . . 6  |-  ( ( x  C_  RR  /\  y  C_  RR )  ->  (
x  X.  y ) 
C_  ( RR  X.  RR ) )
73, 5, 6syl2an 464 . . . . 5  |-  ( ( x  e. 𝔅  /\  y  e. 𝔅 )  ->  ( x  X.  y )  C_  ( RR  X.  RR ) )
8 vex 2904 . . . . . . 7  |-  x  e. 
_V
9 vex 2904 . . . . . . 7  |-  y  e. 
_V
108, 9xpex 4932 . . . . . 6  |-  ( x  X.  y )  e. 
_V
1110elpw 3750 . . . . 5  |-  ( ( x  X.  y )  e.  ~P ( RR 
X.  RR )  <->  ( x  X.  y )  C_  ( RR  X.  RR ) )
127, 11sylibr 204 . . . 4  |-  ( ( x  e. 𝔅  /\  y  e. 𝔅 )  ->  ( x  X.  y )  e.  ~P ( RR  X.  RR ) )
1312rgen2a 2717 . . 3  |-  A. x  e. 𝔅  A. y  e. 𝔅  ( x  X.  y
)  e.  ~P ( RR  X.  RR )
14 eqid 2389 . . . 4  |-  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  =  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )
1514rnmpt2ss 23929 . . 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 8 . 2  |-  ran  (
x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  C_  ~P ( RR  X.  RR )
17 unibrsiga 24338 . . . . . 6  |-  U.𝔅  =  RR
18 brsigarn 24336 . . . . . . 7  |- 𝔅  e.  (sigAlgebra `  RR )
19 elrnsiga 24307 . . . . . . 7  |-  (𝔅  e.  (sigAlgebra `  RR )  -> 𝔅  e.  U. ran sigAlgebra )
20 unielsiga 24309 . . . . . . 7  |-  (𝔅  e.  U. ran sigAlgebra  ->  U.𝔅  e. 𝔅 )
2118, 19, 20mp2b 10 . . . . . 6  |-  U.𝔅  e. 𝔅
2217, 21eqeltrri 2460 . . . . 5  |-  RR  e. 𝔅
23 eqid 2389 . . . . 5  |-  ( RR 
X.  RR )  =  ( RR  X.  RR )
24 xpeq1 4834 . . . . . . 7  |-  ( x  =  RR  ->  (
x  X.  y )  =  ( RR  X.  y ) )
2524eqeq2d 2400 . . . . . 6  |-  ( x  =  RR  ->  (
( RR  X.  RR )  =  ( x  X.  y )  <->  ( RR  X.  RR )  =  ( RR  X.  y ) ) )
26 xpeq2 4835 . . . . . . 7  |-  ( y  =  RR  ->  ( RR  X.  y )  =  ( RR  X.  RR ) )
2726eqeq2d 2400 . . . . . 6  |-  ( y  =  RR  ->  (
( RR  X.  RR )  =  ( RR  X.  y )  <->  ( RR  X.  RR )  =  ( RR  X.  RR ) ) )
2825, 27rspc2ev 3005 . . . . 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 1279 . . . 4  |-  E. x  e. 𝔅  E. y  e. 𝔅  ( RR  X.  RR )  =  ( x  X.  y )
3014, 10elrnmpt2 6124 . . . 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 201 . . 3  |-  ( RR 
X.  RR )  e. 
ran  ( x  e. 𝔅 , 
y  e. 𝔅 
|->  ( x  X.  y
) )
32 elpwuni 4121 . . 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 8 . 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 200 1  |-  U. ran  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  =  ( RR  X.  RR )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   E.wrex 2652    C_ wss 3265   ~Pcpw 3744   U.cuni 3959    X. cxp 4818   ran crn 4821   ` cfv 5396    e. cmpt2 6024   RRcr 8924  sigAlgebracsiga 24288  𝔅cbrsiga 24333
This theorem is referenced by:  sxbrsigalem5  24434  sxbrsiga  24436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-pre-lttri 8999  ax-pre-lttrn 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-ioo 10854  df-topgen 13596  df-top 16888  df-bases 16890  df-siga 24289  df-sigagen 24320  df-brsiga 24334
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