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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 45444 | . . 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 1143 | 1 β’ (π β π· β βͺ π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3062 {crab 3433 β wss 3949 βͺ cuni 4909 class class class wbr 5149 dom cdm 5677 βΆwf 6540 βcfv 6544 (class class class)co 7409 βcr 11109 < clt 11248 βΎt crest 17366 SAlgcsalg 45024 SMblFncsmblfn 45411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-pre-lttri 11184 ax-pre-lttrn 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 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-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 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 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-er 8703 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-ioo 13328 df-ico 13330 df-smblfn 45412 |
This theorem is referenced by: sssmf 45454 smfsssmf 45459 issmfle 45461 smfpimltxr 45463 issmfgt 45472 smfadd 45481 issmfge 45486 smflim 45493 smfpimgtxr 45496 smfpimioo 45503 smfresal 45504 smfrec 45505 smfres 45506 smfmul 45511 smfmulc1 45512 smfco 45518 smfsuplem3 45529 smfpimne 45555 smfpimne2 45556 |
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