![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 45895 | . . 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 3053 {crab 3424 β wss 3940 βͺ cuni 4899 class class class wbr 5138 dom cdm 5666 βΆwf 6529 βcfv 6533 (class class class)co 7401 βcr 11104 < clt 11244 βΎt crest 17362 SAlgcsalg 45475 SMblFncsmblfn 45862 |
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 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-pre-lttri 11179 ax-pre-lttrn 11180 |
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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-er 8698 df-pm 8818 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-ioo 13324 df-ico 13326 df-smblfn 45863 |
This theorem is referenced by: sssmf 45905 smfsssmf 45910 issmfle 45912 smfpimltxr 45914 issmfgt 45923 smfadd 45932 issmfge 45937 smflim 45944 smfpimgtxr 45947 smfpimioo 45954 smfresal 45955 smfrec 45956 smfres 45957 smfmul 45962 smfmulc1 45963 smfco 45969 smfsuplem3 45980 smfpimne 46006 smfpimne2 46007 |
Copyright terms: Public domain | W3C validator |