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Theorem rrvmbfm 34424
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 5917 . . . . 5 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
32oveq1d 7446 . . . 4 (𝑝 = 𝑃 → (dom 𝑝MblFnM𝔅) = (dom 𝑃MblFnM𝔅))
4 df-rrv 34423 . . . 4 rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅))
5 ovex 7464 . . . 4 (dom 𝑃MblFnM𝔅) ∈ V
63, 4, 5fvmpt 7016 . . 3 (𝑃 ∈ Prob → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅))
71, 6syl 17 . 2 (𝜑 → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅))
87eleq2d 2825 1 (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  dom cdm 5689  cfv 6563  (class class class)co 7431  𝔅cbrsiga 34162  MblFnMcmbfm 34230  Probcprb 34389  rRndVarcrrv 34422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-rrv 34423
This theorem is referenced by:  isrrvv  34425  rrvadd  34434  rrvmulc  34435  orrvcval4  34446  orrvcoel  34447  orrvccel  34448
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