Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orrvcoel | Structured version Visualization version GIF version |
Description: If the relation produces open sets, preimage maps of a random variable are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.) |
Ref | Expression |
---|---|
orrvccel.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
orrvccel.2 | ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) |
orrvccel.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
orrvcoel.5 | ⊢ (𝜑 → {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴} ∈ (topGen‘ran (,))) |
Ref | Expression |
---|---|
orrvcoel | ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) ∈ dom 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orrvccel.1 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
2 | domprobsiga 31669 | . . 3 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra) |
4 | retop 23369 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top) |
6 | orrvccel.2 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
7 | 1 | rrvmbfm 31700 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
8 | 6, 7 | mpbid 234 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ)) |
9 | df-brsiga 31441 | . . . 4 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
10 | 9 | oveq2i 7166 | . . 3 ⊢ (dom 𝑃MblFnM𝔅ℝ) = (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,)))) |
11 | 8, 10 | eleqtrdi 2923 | . 2 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,))))) |
12 | orrvccel.4 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
13 | uniretop 23370 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
14 | rabeq 3483 | . . . 4 ⊢ (ℝ = ∪ (topGen‘ran (,)) → {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴} = {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴}) | |
15 | 13, 14 | ax-mp 5 | . . 3 ⊢ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴} = {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴} |
16 | orrvcoel.5 | . . 3 ⊢ (𝜑 → {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴} ∈ (topGen‘ran (,))) | |
17 | 15, 16 | eqeltrrid 2918 | . 2 ⊢ (𝜑 → {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴} ∈ (topGen‘ran (,))) |
18 | 3, 5, 11, 12, 17 | orvcoel 31719 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) ∈ dom 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 ∪ cuni 4837 class class class wbr 5065 dom cdm 5554 ran crn 5555 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 (,)cioo 12737 topGenctg 16710 Topctop 21500 sigAlgebracsiga 31367 sigaGencsigagen 31397 𝔅ℝcbrsiga 31440 MblFnMcmbfm 31508 Probcprb 31665 rRndVarcrrv 31698 ∘RV/𝑐corvc 31713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-pre-lttri 10610 ax-pre-lttrn 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-ioo 12741 df-topgen 16716 df-top 21501 df-bases 21553 df-esum 31287 df-siga 31368 df-sigagen 31398 df-brsiga 31441 df-meas 31455 df-mbfm 31509 df-prob 31666 df-rrv 31699 df-orvc 31714 |
This theorem is referenced by: (None) |
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