Theorem rrvmbfm 34407

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	⊢ (𝜑 → 𝑃 ∈ Prob)

Assertion

Ref	Expression
rrvmbfm	⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ)))

Proof of Theorem rrvmbfm

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isrrvv.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ Prob)
2		dmeq 5928	. . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
3	2	oveq1d 7463	. . . 4 ⊢ (𝑝 = 𝑃 → (dom 𝑝MblFnM𝔅ℝ) = (dom 𝑃MblFnM𝔅ℝ))
4		df-rrv 34406	. . . 4 ⊢ rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅ℝ))
5		ovex 7481	. . . 4 ⊢ (dom 𝑃MblFnM𝔅ℝ) ∈ V
6	3, 4, 5	fvmpt 7029	. . 3 ⊢ (𝑃 ∈ Prob → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅ℝ))
7	1, 6	syl 17	. 2 ⊢ (𝜑 → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅ℝ))
8	7	eleq2d 2830	1 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  dom cdm 5700  ‘cfv 6573  (class class class)co 7448  𝔅_ℝcbrsiga 34145  MblFnMcmbfm 34213  Probcprb 34372  rRndVarcrrv 34405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-rrv 34406

This theorem is referenced by:  isrrvv  34408  rrvadd  34417  rrvmulc  34418  orrvcval4  34429  orrvcoel  34430  orrvccel  34431