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Theorem rrvmbfm 34445
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 (𝜑𝑃 ∈ Prob)
Assertion
Ref Expression
rrvmbfm (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅)))

Proof of Theorem rrvmbfm
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 isrrvv.1 . . 3 (𝜑𝑃 ∈ Prob)
2 dmeq 5913 . . . . 5 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
32oveq1d 7447 . . . 4 (𝑝 = 𝑃 → (dom 𝑝MblFnM𝔅) = (dom 𝑃MblFnM𝔅))
4 df-rrv 34444 . . . 4 rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅))
5 ovex 7465 . . . 4 (dom 𝑃MblFnM𝔅) ∈ V
63, 4, 5fvmpt 7015 . . 3 (𝑃 ∈ Prob → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅))
71, 6syl 17 . 2 (𝜑 → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅))
87eleq2d 2826 1 (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wcel 2107  dom cdm 5684  cfv 6560  (class class class)co 7432  𝔅cbrsiga 34183  MblFnMcmbfm 34251  Probcprb 34410  rRndVarcrrv 34443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-rrv 34444
This theorem is referenced by:  isrrvv  34446  rrvadd  34455  rrvmulc  34456  orrvcval4  34468  orrvcoel  34469  orrvccel  34470
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