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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rrvmbfm | Structured version Visualization version GIF version | ||
| 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.) | 
| Ref | Expression | 
|---|---|
| isrrvv.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) | 
| Ref | Expression | 
|---|---|
| rrvmbfm | ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isrrvv.1 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
| 2 | dmeq 5913 | . . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃) | |
| 3 | 2 | oveq1d 7447 | . . . 4 ⊢ (𝑝 = 𝑃 → (dom 𝑝MblFnM𝔅ℝ) = (dom 𝑃MblFnM𝔅ℝ)) | 
| 4 | df-rrv 34444 | . . . 4 ⊢ rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅ℝ)) | |
| 5 | ovex 7465 | . . . 4 ⊢ (dom 𝑃MblFnM𝔅ℝ) ∈ V | |
| 6 | 3, 4, 5 | fvmpt 7015 | . . 3 ⊢ (𝑃 ∈ Prob → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅ℝ)) | 
| 7 | 1, 6 | syl 17 | . 2 ⊢ (𝜑 → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅ℝ)) | 
| 8 | 7 | eleq2d 2826 | 1 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 dom cdm 5684 ‘cfv 6560 (class class class)co 7432 𝔅ℝcbrsiga 34183 MblFnMcmbfm 34251 Probcprb 34410 rRndVarcrrv 34443 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-rrv 34444 | 
| This theorem is referenced by: isrrvv 34446 rrvadd 34455 rrvmulc 34456 orrvcval4 34468 orrvcoel 34469 orrvccel 34470 | 
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