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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smff | Structured version Visualization version GIF version |
Description: A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smff.s | β’ (π β π β SAlg) |
smff.f | β’ (π β πΉ β (SMblFnβπ)) |
smff.d | β’ π· = dom πΉ |
Ref | Expression |
---|---|
smff | β’ (π β πΉ:π·βΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smff.f | . . 3 β’ (π β πΉ β (SMblFnβπ)) | |
2 | smff.s | . . . 4 β’ (π β π β SAlg) | |
3 | smff.d | . . . 4 β’ π· = dom πΉ | |
4 | 2, 3 | issmf 45954 | . . 3 β’ (π β (πΉ β (SMblFnβπ) β (π· β βͺ π β§ πΉ:π·βΆβ β§ βπ β β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·)))) |
5 | 1, 4 | mpbid 231 | . 2 β’ (π β (π· β βͺ π β§ πΉ:π·βΆβ β§ βπ β β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·))) |
6 | 5 | simp2d 1140 | 1 β’ (π β πΉ:π·βΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3053 {crab 3424 β wss 3941 βͺ cuni 4900 class class class wbr 5139 dom cdm 5667 βΆwf 6530 βcfv 6534 (class class class)co 7402 βcr 11106 < clt 11246 βΎt crest 17367 SAlgcsalg 45534 SMblFncsmblfn 45921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-pre-lttri 11181 ax-pre-lttrn 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-er 8700 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-ioo 13326 df-ico 13328 df-smblfn 45922 |
This theorem is referenced by: sssmf 45964 smfsssmf 45969 issmfle 45971 smfpimltxr 45973 issmfgt 45982 issmfge 45996 smflimlem2 45998 smflimlem3 45999 smflimlem4 46000 smflim 46003 smfpimgtxr 46006 smfpimioompt 46012 smfpimioo 46013 smfresal 46014 smfres 46016 smfco 46028 smffmptf 46030 smfsuplem1 46037 smfsuplem3 46039 smfsupxr 46042 smfinflem 46043 smflimsuplem2 46047 smflimsuplem3 46048 smflimsuplem4 46049 smflimsuplem5 46050 smfliminflem 46056 smfpimne 46065 smfpimne2 46066 smfsupdmmbllem 46070 smfinfdmmbllem 46074 |
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