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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfpreimaltf | 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 |
---|---|
smfpreimaltf.x | β’ β²π₯πΉ |
smfpreimaltf.s | β’ (π β π β SAlg) |
smfpreimaltf.f | β’ (π β πΉ β (SMblFnβπ)) |
smfpreimaltf.d | β’ π· = dom πΉ |
smfpreimaltf.a | β’ (π β π΄ β β) |
Ref | Expression |
---|---|
smfpreimaltf | β’ (π β {π₯ β π· β£ (πΉβπ₯) < π΄} β (π βΎt π·)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfpreimaltf.a | . 2 β’ (π β π΄ β β) | |
2 | smfpreimaltf.f | . . . 4 β’ (π β πΉ β (SMblFnβπ)) | |
3 | smfpreimaltf.x | . . . . 5 β’ β²π₯πΉ | |
4 | smfpreimaltf.s | . . . . 5 β’ (π β π β SAlg) | |
5 | smfpreimaltf.d | . . . . 5 β’ π· = dom πΉ | |
6 | 3, 4, 5 | issmff 46181 | . . . 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·)))) |
7 | 2, 6 | 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·))) |
8 | 7 | simp3d 1141 | . 2 β’ (π β βπ β β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·)) |
9 | breq2 5148 | . . . . 5 β’ (π = π΄ β ((πΉβπ₯) < π β (πΉβπ₯) < π΄)) | |
10 | 9 | rabbidv 3427 | . . . 4 β’ (π = π΄ β {π₯ β π· β£ (πΉβπ₯) < π} = {π₯ β π· β£ (πΉβπ₯) < π΄}) |
11 | 10 | eleq1d 2810 | . . 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·))) |
12 | 11 | rspcva 3601 | . 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·)) |
13 | 1, 8, 12 | syl2anc 582 | 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 1084 = wceq 1533 β wcel 2098 β²wnfc 2875 βwral 3051 {crab 3419 β wss 3941 βͺ cuni 4904 class class class wbr 5144 dom cdm 5673 βΆwf 6539 βcfv 6543 (class class class)co 7413 βcr 11132 < clt 11273 βΎt crest 17396 SAlgcsalg 45755 SMblFncsmblfn 46142 |
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 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-pre-lttri 11207 ax-pre-lttrn 11208 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7987 df-2nd 7988 df-er 8718 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-ioo 13355 df-ico 13357 df-smblfn 46143 |
This theorem is referenced by: smfpimltmpt 46193 smfpimltxr 46194 |
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