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Mathbox for Thierry Arnoux |
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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 33049 | . . . . 5 β’ (π β dom π = βͺ dom π) |
4 | 3 | eleq2d 2824 | . . . 4 β’ (π β (π§ β dom π β π§ β βͺ dom π)) |
5 | 4 | biimprd 248 | . . 3 β’ (π β (π§ β βͺ dom π β π§ β dom π)) |
6 | 5 | imdistani 570 | . 2 β’ ((π β§ π§ β βͺ dom π) β (π β§ π§ β dom π)) |
7 | 1, 2 | rrvfn 33048 | . . . 4 β’ (π β π Fn βͺ dom π) |
8 | fnfun 6603 | . . . 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 33062 | . 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 397 β wcel 2107 βͺ cuni 4866 class class class wbr 5106 dom cdm 5634 Fun wfun 6491 Fn wfn 6492 βcfv 6497 (class class class)co 7358 Probcprb 33010 rRndVarcrrv 33043 βRV/πcorvc 33058 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-pre-lttri 11126 ax-pre-lttrn 11127 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-ioo 13269 df-topgen 17326 df-top 22246 df-bases 22299 df-esum 32630 df-siga 32711 df-sigagen 32741 df-brsiga 32784 df-meas 32798 df-mbfm 32852 df-prob 33011 df-rrv 33044 df-orvc 33059 |
This theorem is referenced by: (None) |
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