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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfpimne2 | Structured version Visualization version GIF version |
Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of reals that are different from a value is in the subspace sigma-algebra induced by its domain. Notice that π΄ is not assumed to be an extended real. (Contributed by Glauco Siliprandi, 5-Jan-2025.) |
Ref | Expression |
---|---|
smfpimne2.p | β’ β²π₯π |
smfpimne2.x | β’ β²π₯πΉ |
smfpimne2.s | β’ (π β π β SAlg) |
smfpimne2.f | β’ (π β πΉ β (SMblFnβπ)) |
smfpimne2.d | β’ π· = dom πΉ |
Ref | Expression |
---|---|
smfpimne2 | β’ (π β {π₯ β π· β£ (πΉβπ₯) β π΄} β (π βΎt π·)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfpimne2.p | . . . 4 β’ β²π₯π | |
2 | nfv 1909 | . . . 4 β’ β²π₯ π΄ β β* | |
3 | 1, 2 | nfan 1894 | . . 3 β’ β²π₯(π β§ π΄ β β*) |
4 | smfpimne2.x | . . 3 β’ β²π₯πΉ | |
5 | smfpimne2.s | . . . 4 β’ (π β π β SAlg) | |
6 | 5 | adantr 479 | . . 3 β’ ((π β§ π΄ β β*) β π β SAlg) |
7 | smfpimne2.f | . . . 4 β’ (π β πΉ β (SMblFnβπ)) | |
8 | 7 | adantr 479 | . . 3 β’ ((π β§ π΄ β β*) β πΉ β (SMblFnβπ)) |
9 | smfpimne2.d | . . 3 β’ π· = dom πΉ | |
10 | simpr 483 | . . 3 β’ ((π β§ π΄ β β*) β π΄ β β*) | |
11 | 3, 4, 6, 8, 9, 10 | smfpimne 46290 | . 2 β’ ((π β§ π΄ β β*) β {π₯ β π· β£ (πΉβπ₯) β π΄} β (π βΎt π·)) |
12 | 4 | nfdm 5947 | . . . . . . 7 β’ β²π₯dom πΉ |
13 | 9, 12 | nfcxfr 2890 | . . . . . 6 β’ β²π₯π· |
14 | 13 | ssrab2f 44548 | . . . . 5 β’ {π₯ β π· β£ (πΉβπ₯) β π΄} β π· |
15 | 14 | a1i 11 | . . . 4 β’ ((π β§ Β¬ π΄ β β*) β {π₯ β π· β£ (πΉβπ₯) β π΄} β π·) |
16 | nfv 1909 | . . . . . 6 β’ β²π₯ Β¬ π΄ β β* | |
17 | 1, 16 | nfan 1894 | . . . . 5 β’ β²π₯(π β§ Β¬ π΄ β β*) |
18 | ssidd 3996 | . . . . 5 β’ ((π β§ Β¬ π΄ β β*) β π· β π·) | |
19 | nne 2934 | . . . . . . . 8 β’ (Β¬ (πΉβπ₯) β π΄ β (πΉβπ₯) = π΄) | |
20 | simpr 483 | . . . . . . . . 9 β’ (((π β§ π₯ β π·) β§ (πΉβπ₯) = π΄) β (πΉβπ₯) = π΄) | |
21 | 5, 7, 9 | smff 46183 | . . . . . . . . . . . 12 β’ (π β πΉ:π·βΆβ) |
22 | 21 | ffvelcdmda 7089 | . . . . . . . . . . 11 β’ ((π β§ π₯ β π·) β (πΉβπ₯) β β) |
23 | 22 | rexrd 11294 | . . . . . . . . . 10 β’ ((π β§ π₯ β π·) β (πΉβπ₯) β β*) |
24 | 23 | adantr 479 | . . . . . . . . 9 β’ (((π β§ π₯ β π·) β§ (πΉβπ₯) = π΄) β (πΉβπ₯) β β*) |
25 | 20, 24 | eqeltrrd 2826 | . . . . . . . 8 β’ (((π β§ π₯ β π·) β§ (πΉβπ₯) = π΄) β π΄ β β*) |
26 | 19, 25 | sylan2b 592 | . . . . . . 7 β’ (((π β§ π₯ β π·) β§ Β¬ (πΉβπ₯) β π΄) β π΄ β β*) |
27 | 26 | adantllr 717 | . . . . . 6 β’ ((((π β§ Β¬ π΄ β β*) β§ π₯ β π·) β§ Β¬ (πΉβπ₯) β π΄) β π΄ β β*) |
28 | simpllr 774 | . . . . . 6 β’ ((((π β§ Β¬ π΄ β β*) β§ π₯ β π·) β§ Β¬ (πΉβπ₯) β π΄) β Β¬ π΄ β β*) | |
29 | 27, 28 | condan 816 | . . . . 5 β’ (((π β§ Β¬ π΄ β β*) β§ π₯ β π·) β (πΉβπ₯) β π΄) |
30 | 13, 13, 17, 18, 29 | ssrabdf 44546 | . . . 4 β’ ((π β§ Β¬ π΄ β β*) β π· β {π₯ β π· β£ (πΉβπ₯) β π΄}) |
31 | 15, 30 | eqssd 3990 | . . 3 β’ ((π β§ Β¬ π΄ β β*) β {π₯ β π· β£ (πΉβπ₯) β π΄} = π·) |
32 | 5, 7, 9 | smfdmss 46184 | . . . . 5 β’ (π β π· β βͺ π) |
33 | 5, 32 | subsaluni 45811 | . . . 4 β’ (π β π· β (π βΎt π·)) |
34 | 33 | adantr 479 | . . 3 β’ ((π β§ Β¬ π΄ β β*) β π· β (π βΎt π·)) |
35 | 31, 34 | eqeltrd 2825 | . 2 β’ ((π β§ Β¬ π΄ β β*) β {π₯ β π· β£ (πΉβπ₯) β π΄} β (π βΎt π·)) |
36 | 11, 35 | pm2.61dan 811 | 1 β’ (π β {π₯ β π· β£ (πΉβπ₯) β π΄} β (π βΎt π·)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 = wceq 1533 β²wnf 1777 β wcel 2098 β²wnfc 2875 β wne 2930 {crab 3419 β wss 3939 dom cdm 5672 βcfv 6543 (class class class)co 7416 βcr 11137 β*cxr 11277 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: smfdivdmmbl2 46292 |
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