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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 33532 | . 2 β’ (π β (πβRV/π E π΄) = (β‘π β {π₯ β β β£ π₯ E π΄})) |
5 | epelg 5581 | . . . . . 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 3440 | . . . 4 β’ (π β {π₯ β β β£ π₯ E π΄} = {π₯ β β β£ π₯ β π΄}) |
8 | dfin5 3956 | . . . . 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 4941 | . . . . . . 7 β’ (π΄ β π β β π΄ β βͺ π β) | |
11 | unibrsiga 33253 | . . . . . . 7 β’ βͺ π β = β | |
12 | 10, 11 | sseqtrdi 4032 | . . . . . 6 β’ (π΄ β π β β π΄ β β) |
13 | 3, 12 | syl 17 | . . . . 5 β’ (π β π΄ β β) |
14 | sseqin2 4215 | . . . . 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 2778 | . . 3 β’ (π β {π₯ β β β£ π₯ E π΄} = π΄) |
17 | 16 | imaeq2d 6059 | . 2 β’ (π β (β‘π β {π₯ β β β£ π₯ E π΄}) = (β‘π β π΄)) |
18 | 4, 17 | eqtrd 2772 | 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 1541 β wcel 2106 {crab 3432 β© cin 3947 β wss 3948 βͺ cuni 4908 class class class wbr 5148 E cep 5579 β‘ccnv 5675 β cima 5679 βcfv 6543 (class class class)co 7411 βcr 11111 π βcbrsiga 33248 Probcprb 33475 rRndVarcrrv 33508 βRV/πcorvc 33523 |
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 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 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-eprel 5580 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 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 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-ioo 13330 df-topgen 17391 df-top 22403 df-bases 22456 df-esum 33095 df-siga 33176 df-sigagen 33206 df-brsiga 33249 df-meas 33263 df-mbfm 33317 df-prob 33476 df-rrv 33509 df-orvc 33524 |
This theorem is referenced by: orvcelel 33537 dstrvval 33538 dstrvprob 33539 |
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