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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orvcelval | Structured version Visualization version GIF version |
Description: Preimage maps produced by the membership relation. (Contributed by Thierry Arnoux, 6-Feb-2017.) |
Ref | Expression |
---|---|
dstrvprob.1 | β’ (π β π β Prob) |
dstrvprob.2 | β’ (π β π β (rRndVarβπ)) |
orvcelel.1 | β’ (π β π΄ β π β) |
Ref | Expression |
---|---|
orvcelval | β’ (π β (πβRV/π E π΄) = (β‘π β π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dstrvprob.1 | . . 3 β’ (π β π β Prob) | |
2 | dstrvprob.2 | . . 3 β’ (π β π β (rRndVarβπ)) | |
3 | orvcelel.1 | . . 3 β’ (π β π΄ β π β) | |
4 | 1, 2, 3 | orrvcval4 33104 | . 2 β’ (π β (πβRV/π E π΄) = (β‘π β {π₯ β β β£ π₯ E π΄})) |
5 | epelg 5543 | . . . . . 6 β’ (π΄ β π β β (π₯ E π΄ β π₯ β π΄)) | |
6 | 3, 5 | syl 17 | . . . . 5 β’ (π β (π₯ E π΄ β π₯ β π΄)) |
7 | 6 | rabbidv 3418 | . . . 4 β’ (π β {π₯ β β β£ π₯ E π΄} = {π₯ β β β£ π₯ β π΄}) |
8 | dfin5 3923 | . . . . 5 β’ (β β© π΄) = {π₯ β β β£ π₯ β π΄} | |
9 | 8 | a1i 11 | . . . 4 β’ (π β (β β© π΄) = {π₯ β β β£ π₯ β π΄}) |
10 | elssuni 4903 | . . . . . . 7 β’ (π΄ β π β β π΄ β βͺ π β) | |
11 | unibrsiga 32825 | . . . . . . 7 β’ βͺ π β = β | |
12 | 10, 11 | sseqtrdi 3999 | . . . . . 6 β’ (π΄ β π β β π΄ β β) |
13 | 3, 12 | syl 17 | . . . . 5 β’ (π β π΄ β β) |
14 | sseqin2 4180 | . . . . 5 β’ (π΄ β β β (β β© π΄) = π΄) | |
15 | 13, 14 | sylib 217 | . . . 4 β’ (π β (β β© π΄) = π΄) |
16 | 7, 9, 15 | 3eqtr2d 2783 | . . 3 β’ (π β {π₯ β β β£ π₯ E π΄} = π΄) |
17 | 16 | imaeq2d 6018 | . 2 β’ (π β (β‘π β {π₯ β β β£ π₯ E π΄}) = (β‘π β π΄)) |
18 | 4, 17 | eqtrd 2777 | 1 β’ (π β (πβRV/π E π΄) = (β‘π β π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 {crab 3410 β© cin 3914 β wss 3915 βͺ cuni 4870 class class class wbr 5110 E cep 5541 β‘ccnv 5637 β cima 5641 βcfv 6501 (class class class)co 7362 βcr 11057 π βcbrsiga 32820 Probcprb 33047 rRndVarcrrv 33080 βRV/πcorvc 33095 |
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 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-pre-lttri 11132 ax-pre-lttrn 11133 |
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 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-ioo 13275 df-topgen 17332 df-top 22259 df-bases 22312 df-esum 32667 df-siga 32748 df-sigagen 32778 df-brsiga 32821 df-meas 32835 df-mbfm 32889 df-prob 33048 df-rrv 33081 df-orvc 33096 |
This theorem is referenced by: orvcelel 33109 dstrvval 33110 dstrvprob 33111 |
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