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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfpimioo | Structured version Visualization version GIF version |
Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfpimioo.s | β’ (π β π β SAlg) |
smfpimioo.f | β’ (π β πΉ β (SMblFnβπ)) |
smfpimioo.d | β’ π· = dom πΉ |
smfpimioo.a | β’ (π β π΄ β β*) |
smfpimioo.b | β’ (π β π΅ β β*) |
Ref | Expression |
---|---|
smfpimioo | β’ (π β (β‘πΉ β (π΄(,)π΅)) β (π βΎt π·)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfpimioo.s | . . . . . . 7 β’ (π β π β SAlg) | |
2 | smfpimioo.f | . . . . . . 7 β’ (π β πΉ β (SMblFnβπ)) | |
3 | smfpimioo.d | . . . . . . 7 β’ π· = dom πΉ | |
4 | 1, 2, 3 | smff 46183 | . . . . . 6 β’ (π β πΉ:π·βΆβ) |
5 | 4 | feqmptd 6962 | . . . . 5 β’ (π β πΉ = (π₯ β π· β¦ (πΉβπ₯))) |
6 | 5 | cnveqd 5872 | . . . 4 β’ (π β β‘πΉ = β‘(π₯ β π· β¦ (πΉβπ₯))) |
7 | 6 | imaeq1d 6057 | . . 3 β’ (π β (β‘πΉ β (π΄(,)π΅)) = (β‘(π₯ β π· β¦ (πΉβπ₯)) β (π΄(,)π΅))) |
8 | eqid 2725 | . . . . 5 β’ (π₯ β π· β¦ (πΉβπ₯)) = (π₯ β π· β¦ (πΉβπ₯)) | |
9 | 8 | mptpreima 6237 | . . . 4 β’ (β‘(π₯ β π· β¦ (πΉβπ₯)) β (π΄(,)π΅)) = {π₯ β π· β£ (πΉβπ₯) β (π΄(,)π΅)} |
10 | 9 | a1i 11 | . . 3 β’ (π β (β‘(π₯ β π· β¦ (πΉβπ₯)) β (π΄(,)π΅)) = {π₯ β π· β£ (πΉβπ₯) β (π΄(,)π΅)}) |
11 | 7, 10 | eqtrd 2765 | . 2 β’ (π β (β‘πΉ β (π΄(,)π΅)) = {π₯ β π· β£ (πΉβπ₯) β (π΄(,)π΅)}) |
12 | nfv 1909 | . . 3 β’ β²π₯π | |
13 | 1 | uniexd 7745 | . . . 4 β’ (π β βͺ π β V) |
14 | 1, 2, 3 | smfdmss 46184 | . . . 4 β’ (π β π· β βͺ π) |
15 | 13, 14 | ssexd 5319 | . . 3 β’ (π β π· β V) |
16 | 4 | ffvelcdmda 7089 | . . 3 β’ ((π β§ π₯ β π·) β (πΉβπ₯) β β) |
17 | 5, 2 | eqeltrrd 2826 | . . 3 β’ (π β (π₯ β π· β¦ (πΉβπ₯)) β (SMblFnβπ)) |
18 | smfpimioo.a | . . 3 β’ (π β π΄ β β*) | |
19 | smfpimioo.b | . . 3 β’ (π β π΅ β β*) | |
20 | 12, 1, 15, 16, 17, 18, 19 | smfpimioompt 46237 | . 2 β’ (π β {π₯ β π· β£ (πΉβπ₯) β (π΄(,)π΅)} β (π βΎt π·)) |
21 | 11, 20 | eqeltrd 2825 | 1 β’ (π β (β‘πΉ β (π΄(,)π΅)) β (π βΎt π·)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {crab 3419 Vcvv 3463 βͺ cuni 4903 β¦ cmpt 5226 β‘ccnv 5671 dom cdm 5672 β cima 5675 βcfv 6543 (class class class)co 7416 βcr 11137 β*cxr 11277 (,)cioo 13356 βΎt crest 17401 SAlgcsalg 45759 SMblFncsmblfn 46146 |
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-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cc 10458 ax-ac2 10486 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
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-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 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-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 df-card 9962 df-acn 9965 df-ac 10139 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-ioo 13360 df-ico 13362 df-fl 13789 df-rest 17403 df-salg 45760 df-smblfn 46147 |
This theorem is referenced by: smfres 46241 smfpimbor1lem1 46249 |
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