Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orrvccel | Structured version Visualization version GIF version |
Description: If the relation produces closed sets, preimage maps 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) |
orrvccel.5 | ⊢ (𝜑 → {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴} ∈ (Clsd‘(topGen‘ran (,)))) |
Ref | Expression |
---|---|
orrvccel | ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) ∈ dom 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orrvccel.1 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
2 | domprobsiga 31671 | . . 3 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra) |
4 | retop 23372 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top) |
6 | orrvccel.2 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
7 | 1 | rrvmbfm 31702 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
8 | 6, 7 | mpbid 234 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ)) |
9 | df-brsiga 31443 | . . . 4 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
10 | 9 | oveq2i 7169 | . . 3 ⊢ (dom 𝑃MblFnM𝔅ℝ) = (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,)))) |
11 | 8, 10 | eleqtrdi 2925 | . 2 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,))))) |
12 | orrvccel.4 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
13 | uniretop 23373 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
14 | rabeq 3485 | . . . 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 | orrvccel.5 | . . 3 ⊢ (𝜑 → {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴} ∈ (Clsd‘(topGen‘ran (,)))) | |
17 | 15, 16 | eqeltrrid 2920 | . 2 ⊢ (𝜑 → {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴} ∈ (Clsd‘(topGen‘ran (,)))) |
18 | 3, 5, 11, 12, 17 | orvccel 31722 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) ∈ dom 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3144 ∪ cuni 4840 class class class wbr 5068 dom cdm 5557 ran crn 5558 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 (,)cioo 12741 topGenctg 16713 Topctop 21503 Clsdccld 21626 sigAlgebracsiga 31369 sigaGencsigagen 31399 𝔅ℝcbrsiga 31442 MblFnMcmbfm 31510 Probcprb 31667 rRndVarcrrv 31700 ∘RV/𝑐corvc 31715 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-ac2 9887 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-oi 8976 df-dju 9332 df-card 9370 df-acn 9373 df-ac 9544 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-ioo 12745 df-topgen 16719 df-top 21504 df-bases 21556 df-cld 21629 df-esum 31289 df-siga 31370 df-sigagen 31400 df-brsiga 31443 df-meas 31457 df-mbfm 31511 df-prob 31668 df-rrv 31701 df-orvc 31716 |
This theorem is referenced by: orvcgteel 31727 orvclteel 31732 |
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