Theorem rrvmbfm 34749

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 5884	. . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
3	2	oveq1d 7415	. . . 4 ⊢ (𝑝 = 𝑃 → (dom 𝑝MblFnM𝔅ℝ) = (dom 𝑃MblFnM𝔅ℝ))
4		df-rrv 34748	. . . 4 ⊢ rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅ℝ))
5		ovex 7433	. . . 4 ⊢ (dom 𝑃MblFnM𝔅ℝ) ∈ V
6	3, 4, 5	fvmpt 6979	. . 3 ⊢ (𝑃 ∈ Prob → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅ℝ))
7	1, 6	syl 18	. 2 ⊢ (𝜑 → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅ℝ))
8	7	eleq2d 2851	1 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1563   ∈ wcel 2145  dom cdm 5652  ‘cfv 6525  (class class class)co 7400  𝔅_ℝcbrsiga 34488  MblFnMcmbfm 34556  Probcprb 34714  rRndVarcrrv 34747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-rrv 34748

This theorem is referenced by:  isrrvv  34750  rrvadd  34759  rrvmulc  34760  orrvcval4  34772  orrvcoel  34773  orrvccel  34774