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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 34413 | . . 3 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra) |
| 4 | retop 24782 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
| 5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top) |
| 6 | orrvccel.2 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
| 7 | 1 | rrvmbfm 34444 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
| 8 | 6, 7 | mpbid 232 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ)) |
| 9 | df-brsiga 34183 | . . . 4 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
| 10 | 9 | oveq2i 7442 | . . 3 ⊢ (dom 𝑃MblFnM𝔅ℝ) = (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,)))) |
| 11 | 8, 10 | eleqtrdi 2851 | . 2 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,))))) |
| 12 | orrvccel.4 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 13 | uniretop 24783 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
| 14 | rabeq 3451 | . . . 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 2846 | . 2 ⊢ (𝜑 → {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴} ∈ (topGen‘ran (,))) |
| 18 | 3, 5, 11, 12, 17 | orvcoel 34464 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) ∈ dom 𝑃) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 ∪ cuni 4907 class class class wbr 5143 dom cdm 5685 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 (,)cioo 13387 topGenctg 17482 Topctop 22899 sigAlgebracsiga 34109 sigaGencsigagen 34139 𝔅ℝcbrsiga 34182 MblFnMcmbfm 34250 Probcprb 34409 rRndVarcrrv 34442 ∘RV/𝑐corvc 34458 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391 df-topgen 17488 df-top 22900 df-bases 22953 df-esum 34029 df-siga 34110 df-sigagen 34140 df-brsiga 34183 df-meas 34197 df-mbfm 34251 df-prob 34410 df-rrv 34443 df-orvc 34459 |
| This theorem is referenced by: (None) |
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