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Theorem br2base 23589
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 23531 . . . . . . . . 9  |- 𝔅 
C_  ~P RR
21sseli 3189 . . . . . . . 8  |-  ( x  e. 𝔅  ->  x  e.  ~P RR )
32elpwid 3647 . . . . . . 7  |-  ( x  e. 𝔅  ->  x  C_  RR )
41sseli 3189 . . . . . . . 8  |-  ( y  e. 𝔅  ->  y  e.  ~P RR )
54elpwid 3647 . . . . . . 7  |-  ( y  e. 𝔅  ->  y  C_  RR )
63, 5anim12i 549 . . . . . 6  |-  ( ( x  e. 𝔅  /\  y  e. 𝔅 )  ->  ( x  C_  RR  /\  y  C_  RR ) )
7 xpss12 4808 . . . . . 6  |-  ( ( x  C_  RR  /\  y  C_  RR )  ->  (
x  X.  y ) 
C_  ( RR  X.  RR ) )
86, 7syl 15 . . . . 5  |-  ( ( x  e. 𝔅  /\  y  e. 𝔅 )  ->  ( x  X.  y )  C_  ( RR  X.  RR ) )
9 vex 2804 . . . . . . 7  |-  x  e. 
_V
10 vex 2804 . . . . . . 7  |-  y  e. 
_V
119, 10xpex 4817 . . . . . 6  |-  ( x  X.  y )  e. 
_V
1211elpw 3644 . . . . 5  |-  ( ( x  X.  y )  e.  ~P ( RR 
X.  RR )  <->  ( x  X.  y )  C_  ( RR  X.  RR ) )
138, 12sylibr 203 . . . 4  |-  ( ( x  e. 𝔅  /\  y  e. 𝔅 )  ->  ( x  X.  y )  e.  ~P ( RR  X.  RR ) )
1413rgen2a 2622 . . 3  |-  A. x  e. 𝔅  A. y  e. 𝔅  ( x  X.  y
)  e.  ~P ( RR  X.  RR )
15 eqid 2296 . . . 4  |-  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  =  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )
1615rnmpt2ss 23254 . . 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 ) )
1714, 16ax-mp 8 . 2  |-  ran  (
x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  C_  ~P ( RR  X.  RR )
18 unibrsiga 23532 . . . . . 6  |-  U.𝔅  =  RR
19 brsigarn 23530 . . . . . . 7  |- 𝔅  e.  (sigAlgebra `  RR )
20 elrnsiga 23502 . . . . . . 7  |-  (𝔅  e.  (sigAlgebra `  RR )  -> 𝔅  e.  U. ran sigAlgebra )
21 unielsiga 23504 . . . . . . 7  |-  (𝔅  e.  U. ran sigAlgebra  ->  U.𝔅  e. 𝔅 )
2219, 20, 21mp2b 9 . . . . . 6  |-  U.𝔅  e. 𝔅
2318, 22eqeltrri 2367 . . . . 5  |-  RR  e. 𝔅
24 eqid 2296 . . . . 5  |-  ( RR 
X.  RR )  =  ( RR  X.  RR )
25 xpeq1 4719 . . . . . . 7  |-  ( x  =  RR  ->  (
x  X.  y )  =  ( RR  X.  y ) )
2625eqeq2d 2307 . . . . . 6  |-  ( x  =  RR  ->  (
( RR  X.  RR )  =  ( x  X.  y )  <->  ( RR  X.  RR )  =  ( RR  X.  y ) ) )
27 xpeq2 4720 . . . . . . 7  |-  ( y  =  RR  ->  ( RR  X.  y )  =  ( RR  X.  RR ) )
2827eqeq2d 2307 . . . . . 6  |-  ( y  =  RR  ->  (
( RR  X.  RR )  =  ( RR  X.  y )  <->  ( RR  X.  RR )  =  ( RR  X.  RR ) ) )
2926, 28rspc2ev 2905 . . . . 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 ) )
3023, 23, 24, 29mp3an 1277 . . . 4  |-  E. x  e. 𝔅  E. y  e. 𝔅  ( RR  X.  RR )  =  ( x  X.  y )
3115, 11elrnmpt2 5973 . . . 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 ) )
3230, 31mpbir 200 . . 3  |-  ( RR 
X.  RR )  e. 
ran  ( x  e. 𝔅 , 
y  e. 𝔅 
|->  ( x  X.  y
) )
33 elpwuni 4005 . . 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 ) ) )
3432, 33ax-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 ) )
3517, 34mpbi 199 1  |-  U. ran  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  =  ( RR  X.  RR )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   ~Pcpw 3638   U.cuni 3843    X. cxp 4703   ran crn 4706   ` cfv 5271    e. cmpt2 5876   RRcr 8752  sigAlgebracsiga 23483  𝔅cbrsiga 23527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-ioo 10676  df-topgen 13360  df-top 16652  df-bases 16654  df-siga 23484  df-sigagen 23515  df-brsiga 23528
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