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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 33410 | . . 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 24278 | . . 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 33441 | . . . 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 33180 | . . . 4 β’ π β = (sigaGenβ(topGenβran (,))) | |
10 | 9 | oveq2i 7420 | . . 3 β’ (dom πMblFnMπ β) = (dom πMblFnM(sigaGenβ(topGenβran (,)))) |
11 | 8, 10 | eleqtrdi 2844 | . 2 β’ (π β π β (dom πMblFnM(sigaGenβ(topGenβran (,))))) |
12 | orrvccel.4 | . 2 β’ (π β π΄ β π) | |
13 | uniretop 24279 | . . . 4 β’ β = βͺ (topGenβran (,)) | |
14 | rabeq 3447 | . . . 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 2839 | . 2 β’ (π β {π¦ β βͺ (topGenβran (,)) β£ π¦π π΄} β (Clsdβ(topGenβran (,)))) |
18 | 3, 5, 11, 12, 17 | orvccel 33461 | 1 β’ (π β (πβRV/ππ π΄) β dom π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {crab 3433 βͺ cuni 4909 class class class wbr 5149 dom cdm 5677 ran crn 5678 βcfv 6544 (class class class)co 7409 βcr 11109 (,)cioo 13324 topGenctg 17383 Topctop 22395 Clsdccld 22520 sigAlgebracsiga 33106 sigaGencsigagen 33136 π βcbrsiga 33179 MblFnMcmbfm 33247 Probcprb 33406 rRndVarcrrv 33439 βRV/πcorvc 33454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-ac2 10458 ax-cnex 11166 ax-resscn 11167 ax-pre-lttri 11184 ax-pre-lttrn 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-oi 9505 df-dju 9896 df-card 9934 df-acn 9937 df-ac 10111 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-ioo 13328 df-topgen 17389 df-top 22396 df-bases 22449 df-cld 22523 df-esum 33026 df-siga 33107 df-sigagen 33137 df-brsiga 33180 df-meas 33194 df-mbfm 33248 df-prob 33407 df-rrv 33440 df-orvc 33455 |
This theorem is referenced by: orvcgteel 33466 orvclteel 33471 |
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