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Theorem rrvmbfm 31707
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 5745 . . . . 5 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
32oveq1d 7145 . . . 4 (𝑝 = 𝑃 → (dom 𝑝MblFnM𝔅) = (dom 𝑃MblFnM𝔅))
4 df-rrv 31706 . . . 4 rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅))
5 ovex 7163 . . . 4 (dom 𝑃MblFnM𝔅) ∈ V
63, 4, 5fvmpt 6741 . . 3 (𝑃 ∈ Prob → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅))
71, 6syl 17 . 2 (𝜑 → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅))
87eleq2d 2897 1 (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115  dom cdm 5528  cfv 6328  (class class class)co 7130  𝔅cbrsiga 31447  MblFnMcmbfm 31515  Probcprb 31672  rRndVarcrrv 31705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-rrv 31706
This theorem is referenced by:  isrrvv  31708  rrvadd  31717  rrvmulc  31718  orrvcval4  31729  orrvcoel  31730  orrvccel  31731
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