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Theorem rrvmbfm 30834
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 5479 . . . . 5 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
32oveq1d 6829 . . . 4 (𝑝 = 𝑃 → (dom 𝑝MblFnM𝔅) = (dom 𝑃MblFnM𝔅))
4 df-rrv 30833 . . . 4 rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅))
5 ovex 6842 . . . 4 (dom 𝑃MblFnM𝔅) ∈ V
63, 4, 5fvmpt 6445 . . 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 196   = wceq 1632  wcel 2139  dom cdm 5266  cfv 6049  (class class class)co 6814  𝔅cbrsiga 30574  MblFnMcmbfm 30642  Probcprb 30799  rRndVarcrrv 30832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-rrv 30833
This theorem is referenced by:  isrrvv  30835  rrvadd  30844  rrvmulc  30845  orrvcval4  30856  orrvcoel  30857  orrvccel  30858
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