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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 33758 | . 2 β’ (π β (πβRV/π E π΄) = (β‘π β {π₯ β β β£ π₯ E π΄})) |
5 | epelg 5582 | . . . . . 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 3439 | . . . 4 β’ (π β {π₯ β β β£ π₯ E π΄} = {π₯ β β β£ π₯ β π΄}) |
8 | dfin5 3957 | . . . . 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 4942 | . . . . . . 7 β’ (π΄ β π β β π΄ β βͺ π β) | |
11 | unibrsiga 33479 | . . . . . . 7 β’ βͺ π β = β | |
12 | 10, 11 | sseqtrdi 4033 | . . . . . 6 β’ (π΄ β π β β π΄ β β) |
13 | 3, 12 | syl 17 | . . . . 5 β’ (π β π΄ β β) |
14 | sseqin2 4216 | . . . . 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 2777 | . . 3 β’ (π β {π₯ β β β£ π₯ E π΄} = π΄) |
17 | 16 | imaeq2d 6060 | . 2 β’ (π β (β‘π β {π₯ β β β£ π₯ E π΄}) = (β‘π β π΄)) |
18 | 4, 17 | eqtrd 2771 | 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 1540 β wcel 2105 {crab 3431 β© cin 3948 β wss 3949 βͺ cuni 4909 class class class wbr 5149 E cep 5580 β‘ccnv 5676 β cima 5680 βcfv 6544 (class class class)co 7412 βcr 11112 π βcbrsiga 33474 Probcprb 33701 rRndVarcrrv 33734 βRV/πcorvc 33749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-pre-lttri 11187 ax-pre-lttrn 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7978 df-2nd 7979 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-ioo 13333 df-topgen 17394 df-top 22617 df-bases 22670 df-esum 33321 df-siga 33402 df-sigagen 33432 df-brsiga 33475 df-meas 33489 df-mbfm 33543 df-prob 33702 df-rrv 33735 df-orvc 33750 |
This theorem is referenced by: orvcelel 33763 dstrvval 33764 dstrvprob 33765 |
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