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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 33438 | . . . . 5 β’ (π β π:βͺ dom πβΆβ) |
4 | dstfrvel.1 | . . . . 5 β’ (π β π΅ β βͺ dom π) | |
5 | 3, 4 | ffvelcdmd 7087 | . . . 4 β’ (π β (πβπ΅) β β) |
6 | dstfrvel.2 | . . . 4 β’ (π β (πβπ΅) β€ π΄) | |
7 | breq1 5151 | . . . . 5 β’ (π₯ = (πβπ΅) β (π₯ β€ π΄ β (πβπ΅) β€ π΄)) | |
8 | 7 | elrab 3683 | . . . 4 β’ ((πβπ΅) β {π₯ β β β£ π₯ β€ π΄} β ((πβπ΅) β β β§ (πβπ΅) β€ π΄)) |
9 | 5, 6, 8 | sylanbrc 583 | . . 3 β’ (π β (πβπ΅) β {π₯ β β β£ π₯ β€ π΄}) |
10 | 3 | ffund 6721 | . . . 4 β’ (π β Fun π) |
11 | 1, 2 | rrvdm 33440 | . . . . 5 β’ (π β dom π = βͺ dom π) |
12 | 4, 11 | eleqtrrd 2836 | . . . 4 β’ (π β π΅ β dom π) |
13 | fvimacnv 7054 | . . . 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 584 | . . 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 33458 | . 2 β’ (π β (πβRV/π β€ π΄) = (β‘π β {π₯ β β β£ π₯ β€ π΄})) |
18 | 15, 17 | eleqtrrd 2836 | 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 2106 {crab 3432 βͺ cuni 4908 class class class wbr 5148 β‘ccnv 5675 dom cdm 5676 β cima 5679 Fun wfun 6537 βcfv 6543 (class class class)co 7408 βcr 11108 β€ cle 11248 Probcprb 33401 rRndVarcrrv 33434 βRV/πcorvc 33449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-ioo 13327 df-topgen 17388 df-top 22395 df-bases 22448 df-esum 33021 df-siga 33102 df-sigagen 33132 df-brsiga 33175 df-meas 33189 df-mbfm 33243 df-prob 33402 df-rrv 33435 df-orvc 33450 |
This theorem is referenced by: dstfrvunirn 33468 |
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