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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 34020 | . 2 β’ (π β (πβRV/π E π΄) = (β‘π β {π₯ β β β£ π₯ E π΄})) |
5 | epelg 5577 | . . . . . 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 3435 | . . . 4 β’ (π β {π₯ β β β£ π₯ E π΄} = {π₯ β β β£ π₯ β π΄}) |
8 | dfin5 3952 | . . . . 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 4935 | . . . . . . 7 β’ (π΄ β π β β π΄ β βͺ π β) | |
11 | unibrsiga 33741 | . . . . . . 7 β’ βͺ π β = β | |
12 | 10, 11 | sseqtrdi 4028 | . . . . . 6 β’ (π΄ β π β β π΄ β β) |
13 | 3, 12 | syl 17 | . . . . 5 β’ (π β π΄ β β) |
14 | sseqin2 4211 | . . . . 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 2773 | . . 3 β’ (π β {π₯ β β β£ π₯ E π΄} = π΄) |
17 | 16 | imaeq2d 6057 | . 2 β’ (π β (β‘π β {π₯ β β β£ π₯ E π΄}) = (β‘π β π΄)) |
18 | 4, 17 | eqtrd 2767 | 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 1534 β wcel 2099 {crab 3427 β© cin 3943 β wss 3944 βͺ cuni 4903 class class class wbr 5142 E cep 5575 β‘ccnv 5671 β cima 5675 βcfv 6542 (class class class)co 7414 βcr 11129 π βcbrsiga 33736 Probcprb 33963 rRndVarcrrv 33996 βRV/πcorvc 34011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-pre-lttri 11204 ax-pre-lttrn 11205 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-ioo 13352 df-topgen 17416 df-top 22783 df-bases 22836 df-esum 33583 df-siga 33664 df-sigagen 33694 df-brsiga 33737 df-meas 33751 df-mbfm 33805 df-prob 33964 df-rrv 33997 df-orvc 34012 |
This theorem is referenced by: orvcelel 34025 dstrvval 34026 dstrvprob 34027 |
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