![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > smfpreimalt | Structured version Visualization version GIF version |
Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfpreimalt.s | β’ (π β π β SAlg) |
smfpreimalt.f | β’ (π β πΉ β (SMblFnβπ)) |
smfpreimalt.d | β’ π· = dom πΉ |
smfpreimalt.a | β’ (π β π΄ β β) |
Ref | Expression |
---|---|
smfpreimalt | β’ (π β {π₯ β π· β£ (πΉβπ₯) < π΄} β (π βΎt π·)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfpreimalt.a | . 2 β’ (π β π΄ β β) | |
2 | smfpreimalt.f | . . . 4 β’ (π β πΉ β (SMblFnβπ)) | |
3 | smfpreimalt.s | . . . . 5 β’ (π β π β SAlg) | |
4 | smfpreimalt.d | . . . . 5 β’ π· = dom πΉ | |
5 | 3, 4 | issmf 46029 | . . . 4 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·)))) |
6 | 2, 5 | mpbid 231 | . . 3 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·))) |
7 | 6 | simp3d 1142 | . 2 β’ (π β βπ β β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·)) |
8 | breq2 5146 | . . . . 5 β’ (π = π΄ β ((πΉβπ₯) < π β (πΉβπ₯) < π΄)) | |
9 | 8 | rabbidv 3435 | . . . 4 β’ (π = π΄ β {π₯ β π· β£ (πΉβπ₯) < π} = {π₯ β π· β£ (πΉβπ₯) < π΄}) |
10 | 9 | eleq1d 2813 | . . 3 β’ (π = π΄ β ({π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·) β {π₯ β π· β£ (πΉβπ₯) < π΄} β (π βΎt π·))) |
11 | 10 | rspcva 3605 | . 2 β’ ((π΄ β β β§ βπ β β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·)) β {π₯ β π· β£ (πΉβπ₯) < π΄} β (π βΎt π·)) |
12 | 1, 7, 11 | syl2anc 583 | 1 β’ (π β {π₯ β π· β£ (πΉβπ₯) < π΄} β (π βΎt π·)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 βwral 3056 {crab 3427 β wss 3944 βͺ cuni 4903 class class class wbr 5142 dom cdm 5672 βΆwf 6538 βcfv 6542 (class class class)co 7414 βcr 11123 < clt 11264 βΎt crest 17387 SAlgcsalg 45609 SMblFncsmblfn 45996 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-pre-lttri 11198 ax-pre-lttrn 11199 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 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 7985 df-2nd 7986 df-er 8716 df-pm 8837 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-ioo 13346 df-ico 13348 df-smblfn 45997 |
This theorem is referenced by: sssmf 46039 smfsssmf 46044 issmfle 46046 smflimlem6 46077 smfco 46103 |
Copyright terms: Public domain | W3C validator |