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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrvfinvima | Structured version Visualization version GIF version |
Description: For a real-value random variable π, any open interval in β is the image of a measurable set. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
isrrvv.1 | β’ (π β π β Prob) |
rrvvf.1 | β’ (π β π β (rRndVarβπ)) |
Ref | Expression |
---|---|
rrvfinvima | β’ (π β βπ¦ β π β (β‘π β π¦) β dom π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrvvf.1 | . . 3 β’ (π β π β (rRndVarβπ)) | |
2 | isrrvv.1 | . . . 4 β’ (π β π β Prob) | |
3 | 2 | isrrvv 34119 | . . 3 β’ (π β (π β (rRndVarβπ) β (π:βͺ dom πβΆβ β§ βπ¦ β π β (β‘π β π¦) β dom π))) |
4 | 1, 3 | mpbid 231 | . 2 β’ (π β (π:βͺ dom πβΆβ β§ βπ¦ β π β (β‘π β π¦) β dom π)) |
5 | 4 | simprd 494 | 1 β’ (π β βπ¦ β π β (β‘π β π¦) β dom π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β wcel 2098 βwral 3051 βͺ cuni 4903 β‘ccnv 5671 dom cdm 5672 β cima 5675 βΆwf 6538 βcfv 6542 βcr 11135 π βcbrsiga 33856 Probcprb 34083 rRndVarcrrv 34116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-pre-lttri 11210 ax-pre-lttrn 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7989 df-2nd 7990 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-ioo 13358 df-topgen 17422 df-top 22812 df-bases 22865 df-esum 33703 df-siga 33784 df-sigagen 33814 df-brsiga 33857 df-meas 33871 df-mbfm 33925 df-prob 34084 df-rrv 34117 |
This theorem is referenced by: orvcelel 34145 dstrvprob 34147 |
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