![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > elorrvc | Structured version Visualization version GIF version |
Description: Elementhood of a preimage for a real-valued random variable. (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) |
Ref | Expression |
---|---|
elorrvc | β’ ((π β§ π§ β βͺ dom π) β (π§ β (πβRV/ππ π΄) β (πβπ§)π π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orrvccel.1 | . . . . . 6 β’ (π β π β Prob) | |
2 | orrvccel.2 | . . . . . 6 β’ (π β π β (rRndVarβπ)) | |
3 | 1, 2 | rrvdm 34071 | . . . . 5 β’ (π β dom π = βͺ dom π) |
4 | 3 | eleq2d 2814 | . . . 4 β’ (π β (π§ β dom π β π§ β βͺ dom π)) |
5 | 4 | biimprd 247 | . . 3 β’ (π β (π§ β βͺ dom π β π§ β dom π)) |
6 | 5 | imdistani 567 | . 2 β’ ((π β§ π§ β βͺ dom π) β (π β§ π§ β dom π)) |
7 | 1, 2 | rrvfn 34070 | . . . 4 β’ (π β π Fn βͺ dom π) |
8 | fnfun 6657 | . . . 4 β’ (π Fn βͺ dom π β Fun π) | |
9 | 7, 8 | syl 17 | . . 3 β’ (π β Fun π) |
10 | orrvccel.4 | . . 3 β’ (π β π΄ β π) | |
11 | 9, 2, 10 | elorvc 34084 | . 2 β’ ((π β§ π§ β dom π) β (π§ β (πβRV/ππ π΄) β (πβπ§)π π΄)) |
12 | 6, 11 | syl 17 | 1 β’ ((π β§ π§ β βͺ dom π) β (π§ β (πβRV/ππ π΄) β (πβπ§)π π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β wcel 2098 βͺ cuni 4910 class class class wbr 5150 dom cdm 5680 Fun wfun 6545 Fn wfn 6546 βcfv 6551 (class class class)co 7424 Probcprb 34032 rRndVarcrrv 34065 βRV/πcorvc 34080 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-pre-lttri 11218 ax-pre-lttrn 11219 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-po 5592 df-so 5593 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 7997 df-2nd 7998 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-ioo 13366 df-topgen 17430 df-top 22814 df-bases 22867 df-esum 33652 df-siga 33733 df-sigagen 33763 df-brsiga 33806 df-meas 33820 df-mbfm 33874 df-prob 34033 df-rrv 34066 df-orvc 34081 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |