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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 34162 | . . 3 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra) |
4 | retop 24722 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top) |
6 | orrvccel.2 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
7 | 1 | rrvmbfm 34193 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
8 | 6, 7 | mpbid 231 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ)) |
9 | df-brsiga 33932 | . . . 4 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
10 | 9 | oveq2i 7430 | . . 3 ⊢ (dom 𝑃MblFnM𝔅ℝ) = (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,)))) |
11 | 8, 10 | eleqtrdi 2835 | . 2 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,))))) |
12 | orrvccel.4 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
13 | uniretop 24723 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
14 | rabeq 3433 | . . . 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 2830 | . 2 ⊢ (𝜑 → {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴} ∈ (topGen‘ran (,))) |
18 | 3, 5, 11, 12, 17 | orvcoel 34212 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) ∈ dom 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3418 ∪ cuni 4909 class class class wbr 5149 dom cdm 5678 ran crn 5679 ‘cfv 6549 (class class class)co 7419 ℝcr 11139 (,)cioo 13359 topGenctg 17422 Topctop 22839 sigAlgebracsiga 33858 sigaGencsigagen 33888 𝔅ℝcbrsiga 33931 MblFnMcmbfm 33999 Probcprb 34158 rRndVarcrrv 34191 ∘RV/𝑐corvc 34206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-pre-lttri 11214 ax-pre-lttrn 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-ioo 13363 df-topgen 17428 df-top 22840 df-bases 22893 df-esum 33778 df-siga 33859 df-sigagen 33889 df-brsiga 33932 df-meas 33946 df-mbfm 34000 df-prob 34159 df-rrv 34192 df-orvc 34207 |
This theorem is referenced by: (None) |
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