Theorem rrvmbfm 31700

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 5772	. . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
3	2	oveq1d 7171	. . . 4 ⊢ (𝑝 = 𝑃 → (dom 𝑝MblFnM𝔅ℝ) = (dom 𝑃MblFnM𝔅ℝ))
4		df-rrv 31699	. . . 4 ⊢ rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅ℝ))
5		ovex 7189	. . . 4 ⊢ (dom 𝑃MblFnM𝔅ℝ) ∈ V
6	3, 4, 5	fvmpt 6768	. . 3 ⊢ (𝑃 ∈ Prob → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅ℝ))
7	1, 6	syl 17	. 2 ⊢ (𝜑 → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅ℝ))
8	7	eleq2d 2898	1 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ)))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114  dom cdm 5555  ‘cfv 6355  (class class class)co 7156  𝔅_ℝcbrsiga 31440  MblFnMcmbfm 31508  Probcprb 31665  rRndVarcrrv 31698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-ov 7159  df-rrv 31699

This theorem is referenced by:  isrrvv  31701  rrvadd  31710  rrvmulc  31711  orrvcval4  31722  orrvcoel  31723  orrvccel  31724