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Theorem br2base 26684
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 26599 . . . . . . . 8  |- 𝔅 
C_  ~P RR
21sseli 3352 . . . . . . 7  |-  ( x  e. 𝔅  ->  x  e.  ~P RR )
32elpwid 3870 . . . . . 6  |-  ( x  e. 𝔅  ->  x  C_  RR )
41sseli 3352 . . . . . . 7  |-  ( y  e. 𝔅  ->  y  e.  ~P RR )
54elpwid 3870 . . . . . 6  |-  ( y  e. 𝔅  ->  y  C_  RR )
6 xpss12 4945 . . . . . 6  |-  ( ( x  C_  RR  /\  y  C_  RR )  ->  (
x  X.  y ) 
C_  ( RR  X.  RR ) )
73, 5, 6syl2an 477 . . . . 5  |-  ( ( x  e. 𝔅  /\  y  e. 𝔅 )  ->  ( x  X.  y )  C_  ( RR  X.  RR ) )
8 vex 2975 . . . . . . 7  |-  x  e. 
_V
9 vex 2975 . . . . . . 7  |-  y  e. 
_V
108, 9xpex 6508 . . . . . 6  |-  ( x  X.  y )  e. 
_V
1110elpw 3866 . . . . 5  |-  ( ( x  X.  y )  e.  ~P ( RR 
X.  RR )  <->  ( x  X.  y )  C_  ( RR  X.  RR ) )
127, 11sylibr 212 . . . 4  |-  ( ( x  e. 𝔅  /\  y  e. 𝔅 )  ->  ( x  X.  y )  e.  ~P ( RR  X.  RR ) )
1312rgen2a 2782 . . 3  |-  A. x  e. 𝔅  A. y  e. 𝔅  ( x  X.  y
)  e.  ~P ( RR  X.  RR )
14 eqid 2443 . . . 4  |-  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  =  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )
1514rnmpt2ss 25992 . . 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 26600 . . . . . 6  |-  U.𝔅  =  RR
18 brsigarn 26598 . . . . . . 7  |- 𝔅  e.  (sigAlgebra `  RR )
19 elrnsiga 26569 . . . . . . 7  |-  (𝔅  e.  (sigAlgebra `  RR )  -> 𝔅  e.  U. ran sigAlgebra )
20 unielsiga 26571 . . . . . . 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 2443 . . . . 5  |-  ( RR 
X.  RR )  =  ( RR  X.  RR )
24 xpeq1 4854 . . . . . . 7  |-  ( x  =  RR  ->  (
x  X.  y )  =  ( RR  X.  y ) )
2524eqeq2d 2454 . . . . . 6  |-  ( x  =  RR  ->  (
( RR  X.  RR )  =  ( x  X.  y )  <->  ( RR  X.  RR )  =  ( RR  X.  y ) ) )
26 xpeq2 4855 . . . . . . 7  |-  ( y  =  RR  ->  ( RR  X.  y )  =  ( RR  X.  RR ) )
2726eqeq2d 2454 . . . . . 6  |-  ( y  =  RR  ->  (
( RR  X.  RR )  =  ( RR  X.  y )  <->  ( RR  X.  RR )  =  ( RR  X.  RR ) ) )
2825, 27rspc2ev 3081 . . . . 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 1314 . . . 4  |-  E. x  e. 𝔅  E. y  e. 𝔅  ( RR  X.  RR )  =  ( x  X.  y )
3014, 10elrnmpt2 6203 . . . 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 209 . . 3  |-  ( RR 
X.  RR )  e. 
ran  ( x  e. 𝔅 , 
y  e. 𝔅 
|->  ( x  X.  y
) )
32 elpwuni 4258 . . 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 208 1  |-  U. ran  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  =  ( RR  X.  RR )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716    C_ wss 3328   ~Pcpw 3860   U.cuni 4091    X. cxp 4838   ran crn 4841   ` cfv 5418    e. cmpt2 6093   RRcr 9281  sigAlgebracsiga 26550  𝔅cbrsiga 26595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-pre-lttri 9356  ax-pre-lttrn 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-ioo 11304  df-topgen 14382  df-top 18503  df-bases 18505  df-siga 26551  df-sigagen 26582  df-brsiga 26596
This theorem is referenced by:  sxbrsigalem5  26703  sxbrsiga  26705
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