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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 45055 | . . 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 1144 | 1 β’ (π β πΉ:π·βΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 {crab 3406 β wss 3911 βͺ cuni 4866 class class class wbr 5106 dom cdm 5634 βΆwf 6493 βcfv 6497 (class class class)co 7358 βcr 11055 < clt 11194 βΎt crest 17307 SAlgcsalg 44635 SMblFncsmblfn 45022 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-pre-lttri 11130 ax-pre-lttrn 11131 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8651 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-ioo 13274 df-ico 13276 df-smblfn 45023 |
This theorem is referenced by: sssmf 45065 smfsssmf 45070 issmfle 45072 smfpimltxr 45074 issmfgt 45083 issmfge 45097 smflimlem2 45099 smflimlem3 45100 smflimlem4 45101 smflim 45104 smfpimgtxr 45107 smfpimioompt 45113 smfpimioo 45114 smfresal 45115 smfres 45117 smfco 45129 smffmptf 45131 smfsuplem1 45138 smfsuplem3 45140 smfsupxr 45143 smfinflem 45144 smflimsuplem2 45148 smflimsuplem3 45149 smflimsuplem4 45150 smflimsuplem5 45151 smfliminflem 45157 smfpimne 45166 smfpimne2 45167 smfsupdmmbllem 45171 smfinfdmmbllem 45175 |
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