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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 46110 | . . 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 1141 | 1 β’ (π β πΉ:π·βΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 βwral 3057 {crab 3428 β wss 3945 βͺ cuni 4903 class class class wbr 5142 dom cdm 5672 βΆwf 6538 βcfv 6542 (class class class)co 7414 βcr 11131 < clt 11272 βΎt crest 17395 SAlgcsalg 45690 SMblFncsmblfn 46077 |
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 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-pre-lttri 11206 ax-pre-lttrn 11207 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 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-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-ioo 13354 df-ico 13356 df-smblfn 46078 |
This theorem is referenced by: sssmf 46120 smfsssmf 46125 issmfle 46127 smfpimltxr 46129 issmfgt 46138 issmfge 46152 smflimlem2 46154 smflimlem3 46155 smflimlem4 46156 smflim 46159 smfpimgtxr 46162 smfpimioompt 46168 smfpimioo 46169 smfresal 46170 smfres 46172 smfco 46184 smffmptf 46186 smfsuplem1 46193 smfsuplem3 46195 smfsupxr 46198 smfinflem 46199 smflimsuplem2 46203 smflimsuplem3 46204 smflimsuplem4 46205 smflimsuplem5 46206 smfliminflem 46212 smfpimne 46221 smfpimne2 46222 smfsupdmmbllem 46226 smfinfdmmbllem 46230 |
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