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Theorem rrvmbfm 28873
Description: A real-valued random variable is a measurable function from its sample space to the Borel Sigma Algebra. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Hypothesis
Ref Expression
isrrvv.1  |-  ( ph  ->  P  e. Prob )
Assertion
Ref Expression
rrvmbfm  |-  ( ph  ->  ( X  e.  (rRndVar `  P )  <->  X  e.  ( dom  PMblFnM𝔅 ) ) )

Proof of Theorem rrvmbfm
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 isrrvv.1 . . 3  |-  ( ph  ->  P  e. Prob )
2 dmeq 5023 . . . . 5  |-  ( p  =  P  ->  dom  p  =  dom  P )
32oveq1d 6292 . . . 4  |-  ( p  =  P  ->  ( dom  pMblFnM𝔅 )  =  ( dom  PMblFnM𝔅 )
)
4 df-rrv 28872 . . . 4  |- rRndVar  =  ( p  e. Prob  |->  ( dom  pMblFnM𝔅 ) )
5 ovex 6305 . . . 4  |-  ( dom 
PMblFnM𝔅 )  e.  _V
63, 4, 5fvmpt 5931 . . 3  |-  ( P  e. Prob  ->  (rRndVar `  P )  =  ( dom  PMblFnM𝔅 ) )
71, 6syl 17 . 2  |-  ( ph  ->  (rRndVar `  P )  =  ( dom  PMblFnM𝔅 ) )
87eleq2d 2472 1  |-  ( ph  ->  ( X  e.  (rRndVar `  P )  <->  X  e.  ( dom  PMblFnM𝔅 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1405    e. wcel 1842   dom cdm 4822   ` cfv 5568  (class class class)co 6277  𝔅cbrsiga 28615  MblFnMcmbfm 28684  Probcprb 28838  rRndVarcrrv 28871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-ov 6280  df-rrv 28872
This theorem is referenced by:  isrrvv  28874  rrvadd  28883  rrvmulc  28884  orrvcval4  28895  orrvcoel  28896  orrvccel  28897
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