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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfdmss | Structured version Visualization version GIF version |
Description: The domain of a function measurable w.r.t. to a sigma-algebra, is a subset of the set underlying the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfdmss.s | β’ (π β π β SAlg) |
smfdmss.f | β’ (π β πΉ β (SMblFnβπ)) |
smfdmss.d | β’ π· = dom πΉ |
Ref | Expression |
---|---|
smfdmss | β’ (π β π· β βͺ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfdmss.f | . . 3 β’ (π β πΉ β (SMblFnβπ)) | |
2 | smfdmss.s | . . . 4 β’ (π β π β SAlg) | |
3 | smfdmss.d | . . . 4 β’ π· = dom πΉ | |
4 | 2, 3 | issmf 46145 | . . 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 | simp1d 1139 | 1 β’ (π β π· β βͺ π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3058 {crab 3430 β wss 3949 βͺ cuni 4912 class class class wbr 5152 dom cdm 5682 βΆwf 6549 βcfv 6553 (class class class)co 7426 βcr 11145 < clt 11286 βΎt crest 17409 SAlgcsalg 45725 SMblFncsmblfn 46112 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-pre-lttri 11220 ax-pre-lttrn 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-er 8731 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-ioo 13368 df-ico 13370 df-smblfn 46113 |
This theorem is referenced by: sssmf 46155 smfsssmf 46160 issmfle 46162 smfpimltxr 46164 issmfgt 46173 smfadd 46182 issmfge 46187 smflim 46194 smfpimgtxr 46197 smfpimioo 46204 smfresal 46205 smfrec 46206 smfres 46207 smfmul 46212 smfmulc1 46213 smfco 46219 smfsuplem3 46230 smfpimne 46256 smfpimne2 46257 |
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