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 31662 | . . 3 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra) |
4 | retop 23362 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top) |
6 | orrvccel.2 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
7 | 1 | rrvmbfm 31693 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
8 | 6, 7 | mpbid 234 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ)) |
9 | df-brsiga 31434 | . . . 4 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
10 | 9 | oveq2i 7159 | . . 3 ⊢ (dom 𝑃MblFnM𝔅ℝ) = (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,)))) |
11 | 8, 10 | eleqtrdi 2921 | . 2 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,))))) |
12 | orrvccel.4 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
13 | uniretop 23363 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
14 | rabeq 3482 | . . . 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 2916 | . 2 ⊢ (𝜑 → {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴} ∈ (Clsd‘(topGen‘ran (,)))) |
18 | 3, 5, 11, 12, 17 | orvccel 31713 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) ∈ dom 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 {crab 3140 ∪ cuni 4830 class class class wbr 5057 dom cdm 5548 ran crn 5549 ‘cfv 6348 (class class class)co 7148 ℝcr 10528 (,)cioo 12730 topGenctg 16703 Topctop 21493 Clsdccld 21616 sigAlgebracsiga 31360 sigaGencsigagen 31390 𝔅ℝcbrsiga 31433 MblFnMcmbfm 31501 Probcprb 31658 rRndVarcrrv 31691 ∘RV/𝑐corvc 31706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-ac2 9877 ax-cnex 10585 ax-resscn 10586 ax-pre-lttri 10603 ax-pre-lttrn 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-oi 8966 df-dju 9322 df-card 9360 df-acn 9363 df-ac 9534 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-ioo 12734 df-topgen 16709 df-top 21494 df-bases 21546 df-cld 21619 df-esum 31280 df-siga 31361 df-sigagen 31391 df-brsiga 31434 df-meas 31448 df-mbfm 31502 df-prob 31659 df-rrv 31692 df-orvc 31707 |
This theorem is referenced by: orvcgteel 31718 orvclteel 31723 |
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