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Theorem rrvmbfm 34289
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 5902 . . . . 5 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
32oveq1d 7431 . . . 4 (𝑝 = 𝑃 → (dom 𝑝MblFnM𝔅) = (dom 𝑃MblFnM𝔅))
4 df-rrv 34288 . . . 4 rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅))
5 ovex 7449 . . . 4 (dom 𝑃MblFnM𝔅) ∈ V
63, 4, 5fvmpt 7001 . . 3 (𝑃 ∈ Prob → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅))
71, 6syl 17 . 2 (𝜑 → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅))
87eleq2d 2812 1 (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  dom cdm 5674  cfv 6546  (class class class)co 7416  𝔅cbrsiga 34027  MblFnMcmbfm 34095  Probcprb 34254  rRndVarcrrv 34287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-iota 6498  df-fun 6548  df-fv 6554  df-ov 7419  df-rrv 34288
This theorem is referenced by:  isrrvv  34290  rrvadd  34299  rrvmulc  34300  orrvcval4  34311  orrvcoel  34312  orrvccel  34313
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