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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dstfrvel | Structured version Visualization version GIF version |
Description: Elementhood of preimage maps produced by the "less than or equal to" relation. (Contributed by Thierry Arnoux, 13-Feb-2017.) |
Ref | Expression |
---|---|
dstfrv.1 | β’ (π β π β Prob) |
dstfrv.2 | β’ (π β π β (rRndVarβπ)) |
orvclteel.1 | β’ (π β π΄ β β) |
dstfrvel.1 | β’ (π β π΅ β βͺ dom π) |
dstfrvel.2 | β’ (π β (πβπ΅) β€ π΄) |
Ref | Expression |
---|---|
dstfrvel | β’ (π β π΅ β (πβRV/π β€ π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dstfrv.1 | . . . . . 6 β’ (π β π β Prob) | |
2 | dstfrv.2 | . . . . . 6 β’ (π β π β (rRndVarβπ)) | |
3 | 1, 2 | rrvvf 33741 | . . . . 5 β’ (π β π:βͺ dom πβΆβ) |
4 | dstfrvel.1 | . . . . 5 β’ (π β π΅ β βͺ dom π) | |
5 | 3, 4 | ffvelcdmd 7086 | . . . 4 β’ (π β (πβπ΅) β β) |
6 | dstfrvel.2 | . . . 4 β’ (π β (πβπ΅) β€ π΄) | |
7 | breq1 5150 | . . . . 5 β’ (π₯ = (πβπ΅) β (π₯ β€ π΄ β (πβπ΅) β€ π΄)) | |
8 | 7 | elrab 3682 | . . . 4 β’ ((πβπ΅) β {π₯ β β β£ π₯ β€ π΄} β ((πβπ΅) β β β§ (πβπ΅) β€ π΄)) |
9 | 5, 6, 8 | sylanbrc 581 | . . 3 β’ (π β (πβπ΅) β {π₯ β β β£ π₯ β€ π΄}) |
10 | 3 | ffund 6720 | . . . 4 β’ (π β Fun π) |
11 | 1, 2 | rrvdm 33743 | . . . . 5 β’ (π β dom π = βͺ dom π) |
12 | 4, 11 | eleqtrrd 2834 | . . . 4 β’ (π β π΅ β dom π) |
13 | fvimacnv 7053 | . . . 4 β’ ((Fun π β§ π΅ β dom π) β ((πβπ΅) β {π₯ β β β£ π₯ β€ π΄} β π΅ β (β‘π β {π₯ β β β£ π₯ β€ π΄}))) | |
14 | 10, 12, 13 | syl2anc 582 | . . 3 β’ (π β ((πβπ΅) β {π₯ β β β£ π₯ β€ π΄} β π΅ β (β‘π β {π₯ β β β£ π₯ β€ π΄}))) |
15 | 9, 14 | mpbid 231 | . 2 β’ (π β π΅ β (β‘π β {π₯ β β β£ π₯ β€ π΄})) |
16 | orvclteel.1 | . . 3 β’ (π β π΄ β β) | |
17 | 1, 2, 16 | orrvcval4 33761 | . 2 β’ (π β (πβRV/π β€ π΄) = (β‘π β {π₯ β β β£ π₯ β€ π΄})) |
18 | 15, 17 | eleqtrrd 2834 | 1 β’ (π β π΅ β (πβRV/π β€ π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β wcel 2104 {crab 3430 βͺ cuni 4907 class class class wbr 5147 β‘ccnv 5674 dom cdm 5675 β cima 5678 Fun wfun 6536 βcfv 6542 (class class class)co 7411 βcr 11111 β€ cle 11253 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-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 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-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 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-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 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-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 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 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-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: dstfrvunirn 33771 |
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