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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 | β’ (π β π β Prob) |
orrvccel.2 | β’ (π β π β (rRndVarβπ)) |
orrvccel.4 | β’ (π β π΄ β π) |
orrvccel.5 | β’ (π β {π¦ β β β£ π¦π π΄} β (Clsdβ(topGenβran (,)))) |
Ref | Expression |
---|---|
orrvccel | β’ (π β (πβRV/ππ π΄) β dom π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orrvccel.1 | . . 3 β’ (π β π β Prob) | |
2 | domprobsiga 33708 | . . 3 β’ (π β Prob β dom π β βͺ ran sigAlgebra) | |
3 | 1, 2 | syl 17 | . 2 β’ (π β dom π β βͺ ran sigAlgebra) |
4 | retop 24498 | . . 3 β’ (topGenβran (,)) β Top | |
5 | 4 | a1i 11 | . 2 β’ (π β (topGenβran (,)) β Top) |
6 | orrvccel.2 | . . . 4 β’ (π β π β (rRndVarβπ)) | |
7 | 1 | rrvmbfm 33739 | . . . 4 β’ (π β (π β (rRndVarβπ) β π β (dom πMblFnMπ β))) |
8 | 6, 7 | mpbid 231 | . . 3 β’ (π β π β (dom πMblFnMπ β)) |
9 | df-brsiga 33478 | . . . 4 β’ π β = (sigaGenβ(topGenβran (,))) | |
10 | 9 | oveq2i 7422 | . . 3 β’ (dom πMblFnMπ β) = (dom πMblFnM(sigaGenβ(topGenβran (,)))) |
11 | 8, 10 | eleqtrdi 2841 | . 2 β’ (π β π β (dom πMblFnM(sigaGenβ(topGenβran (,))))) |
12 | orrvccel.4 | . 2 β’ (π β π΄ β π) | |
13 | uniretop 24499 | . . . 4 β’ β = βͺ (topGenβran (,)) | |
14 | rabeq 3444 | . . . 4 β’ (β = βͺ (topGenβran (,)) β {π¦ β β β£ π¦π π΄} = {π¦ β βͺ (topGenβran (,)) β£ π¦π π΄}) | |
15 | 13, 14 | ax-mp 5 | . . 3 β’ {π¦ β β β£ π¦π π΄} = {π¦ β βͺ (topGenβran (,)) β£ π¦π π΄} |
16 | orrvccel.5 | . . 3 β’ (π β {π¦ β β β£ π¦π π΄} β (Clsdβ(topGenβran (,)))) | |
17 | 15, 16 | eqeltrrid 2836 | . 2 β’ (π β {π¦ β βͺ (topGenβran (,)) β£ π¦π π΄} β (Clsdβ(topGenβran (,)))) |
18 | 3, 5, 11, 12, 17 | orvccel 33759 | 1 β’ (π β (πβRV/ππ π΄) β dom π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 {crab 3430 βͺ cuni 4907 class class class wbr 5147 dom cdm 5675 ran crn 5676 βcfv 6542 (class class class)co 7411 βcr 11111 (,)cioo 13328 topGenctg 17387 Topctop 22615 Clsdccld 22740 sigAlgebracsiga 33404 sigaGencsigagen 33434 π βcbrsiga 33477 MblFnMcmbfm 33545 Probcprb 33704 rRndVarcrrv 33737 βRV/πcorvc 33752 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-ac 10113 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-ioo 13332 df-topgen 17393 df-top 22616 df-bases 22669 df-cld 22743 df-esum 33324 df-siga 33405 df-sigagen 33435 df-brsiga 33478 df-meas 33492 df-mbfm 33546 df-prob 33705 df-rrv 33738 df-orvc 33753 |
This theorem is referenced by: orvcgteel 33764 orvclteel 33769 |
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