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Theorem orrvcval4 28578
Description: The value of the preimage mapping operator can be restricted to preimages of subsets of RR. (Contributed by Thierry Arnoux, 21-Jan-2017.)
Hypotheses
Ref Expression
orrvccel.1  |-  ( ph  ->  P  e. Prob )
orrvccel.2  |-  ( ph  ->  X  e.  (rRndVar `  P
) )
orrvccel.4  |-  ( ph  ->  A  e.  V )
Assertion
Ref Expression
orrvcval4  |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  e.  RR  |  y R A } ) )
Distinct variable groups:    y, A    y, R    y, X
Allowed substitution hints:    ph( y)    P( y)    V( y)

Proof of Theorem orrvcval4
StepHypRef Expression
1 orrvccel.1 . . . 4  |-  ( ph  ->  P  e. Prob )
2 domprobsiga 28525 . . . 4  |-  ( P  e. Prob  ->  dom  P  e.  U.
ran sigAlgebra )
31, 2syl 16 . . 3  |-  ( ph  ->  dom  P  e.  U. ran sigAlgebra )
4 retop 21393 . . . 4  |-  ( topGen ` 
ran  (,) )  e.  Top
54a1i 11 . . 3  |-  ( ph  ->  ( topGen `  ran  (,) )  e.  Top )
6 orrvccel.2 . . . . 5  |-  ( ph  ->  X  e.  (rRndVar `  P
) )
71rrvmbfm 28556 . . . . 5  |-  ( ph  ->  ( X  e.  (rRndVar `  P )  <->  X  e.  ( dom  PMblFnM𝔅 ) ) )
86, 7mpbid 210 . . . 4  |-  ( ph  ->  X  e.  ( dom 
PMblFnM𝔅 ) )
9 df-brsiga 28314 . . . . 5  |- 𝔅  =  (sigaGen `  ( topGen `
 ran  (,) )
)
109oveq2i 6307 . . . 4  |-  ( dom 
PMblFnM𝔅 )  =  ( dom  PMblFnM (sigaGen `  ( topGen `  ran  (,) ) ) )
118, 10syl6eleq 2555 . . 3  |-  ( ph  ->  X  e.  ( dom 
PMblFnM (sigaGen `  ( topGen ` 
ran  (,) ) ) ) )
12 orrvccel.4 . . 3  |-  ( ph  ->  A  e.  V )
133, 5, 11, 12orvcval4 28574 . 2  |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  e.  U. ( topGen ` 
ran  (,) )  |  y R A } ) )
14 uniretop 21394 . . . 4  |-  RR  =  U. ( topGen `  ran  (,) )
15 rabeq 3103 . . . 4  |-  ( RR  =  U. ( topGen ` 
ran  (,) )  ->  { y  e.  RR  |  y R A }  =  { y  e.  U. ( topGen `  ran  (,) )  |  y R A } )
1614, 15ax-mp 5 . . 3  |-  { y  e.  RR  |  y R A }  =  { y  e.  U. ( topGen `  ran  (,) )  |  y R A }
1716imaeq2i 5345 . 2  |-  ( `' X " { y  e.  RR  |  y R A } )  =  ( `' X " { y  e.  U. ( topGen `  ran  (,) )  |  y R A } )
1813, 17syl6eqr 2516 1  |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  e.  RR  |  y R A } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   {crab 2811   U.cuni 4251   class class class wbr 4456   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011   ` cfv 5594  (class class class)co 6296   RRcr 9508   (,)cioo 11554   topGenctg 14854   Topctop 19520  sigAlgebracsiga 28268  sigaGencsigagen 28299  𝔅cbrsiga 28313  MblFnMcmbfm 28382  Probcprb 28521  rRndVarcrrv 28554  ∘RV/𝑐corvc 28569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-ioo 11558  df-topgen 14860  df-top 19525  df-bases 19527  df-esum 28194  df-siga 28269  df-sigagen 28300  df-brsiga 28314  df-meas 28328  df-mbfm 28383  df-prob 28522  df-rrv 28555  df-orvc 28570
This theorem is referenced by:  orvcelval  28582  dstfrvel  28587  orvclteinc  28589
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