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Theorem br2base 27991
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 27907 . . . . . . . 8  |- 𝔅 
C_  ~P RR
21sseli 3500 . . . . . . 7  |-  ( x  e. 𝔅  ->  x  e.  ~P RR )
32elpwid 4020 . . . . . 6  |-  ( x  e. 𝔅  ->  x  C_  RR )
41sseli 3500 . . . . . . 7  |-  ( y  e. 𝔅  ->  y  e.  ~P RR )
54elpwid 4020 . . . . . 6  |-  ( y  e. 𝔅  ->  y  C_  RR )
6 xpss12 5108 . . . . . 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 3116 . . . . . . 7  |-  x  e. 
_V
9 vex 3116 . . . . . . 7  |-  y  e. 
_V
108, 9xpex 6589 . . . . . 6  |-  ( x  X.  y )  e. 
_V
1110elpw 4016 . . . . 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 2891 . . 3  |-  A. x  e. 𝔅  A. y  e. 𝔅  ( x  X.  y
)  e.  ~P ( RR  X.  RR )
14 eqid 2467 . . . 4  |-  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )  =  ( x  e. 𝔅 ,  y  e. 𝔅 
|->  ( x  X.  y
) )
1514rnmpt2ss 27284 . . 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 27908 . . . . . 6  |-  U.𝔅  =  RR
18 brsigarn 27906 . . . . . . 7  |- 𝔅  e.  (sigAlgebra `  RR )
19 elrnsiga 27877 . . . . . . 7  |-  (𝔅  e.  (sigAlgebra `  RR )  -> 𝔅  e.  U. ran sigAlgebra )
20 unielsiga 27879 . . . . . . 7  |-  (𝔅  e.  U. ran sigAlgebra  ->  U.𝔅  e. 𝔅 )
2118, 19, 20mp2b 10 . . . . . 6  |-  U.𝔅  e. 𝔅
2217, 21eqeltrri 2552 . . . . 5  |-  RR  e. 𝔅
23 eqid 2467 . . . . 5  |-  ( RR 
X.  RR )  =  ( RR  X.  RR )
24 xpeq1 5013 . . . . . . 7  |-  ( x  =  RR  ->  (
x  X.  y )  =  ( RR  X.  y ) )
2524eqeq2d 2481 . . . . . 6  |-  ( x  =  RR  ->  (
( RR  X.  RR )  =  ( x  X.  y )  <->  ( RR  X.  RR )  =  ( RR  X.  y ) ) )
26 xpeq2 5014 . . . . . . 7  |-  ( y  =  RR  ->  ( RR  X.  y )  =  ( RR  X.  RR ) )
2726eqeq2d 2481 . . . . . 6  |-  ( y  =  RR  ->  (
( RR  X.  RR )  =  ( RR  X.  y )  <->  ( RR  X.  RR )  =  ( RR  X.  RR ) ) )
2825, 27rspc2ev 3225 . . . . 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 1324 . . . 4  |-  E. x  e. 𝔅  E. y  e. 𝔅  ( RR  X.  RR )  =  ( x  X.  y )
3014, 10elrnmpt2 6400 . . . 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 4413 . . 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 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   ~Pcpw 4010   U.cuni 4245    X. cxp 4997   ran crn 5000   ` cfv 5588    |-> cmpt2 6287   RRcr 9492  sigAlgebracsiga 27858  𝔅cbrsiga 27903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-pre-lttri 9567  ax-pre-lttrn 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-ioo 11534  df-topgen 14702  df-top 19206  df-bases 19208  df-siga 27859  df-sigagen 27890  df-brsiga 27904
This theorem is referenced by:  sxbrsigalem5  28010  sxbrsiga  28012
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