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Mirrors > Home > MPE Home > Th. List > mbfdm | Structured version Visualization version GIF version |
Description: The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
Ref | Expression |
---|---|
mbfdm | β’ (πΉ β MblFn β dom πΉ β dom vol) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ref 15055 | . . . 4 β’ β:ββΆβ | |
2 | mbff 25133 | . . . 4 β’ (πΉ β MblFn β πΉ:dom πΉβΆβ) | |
3 | fco 6738 | . . . 4 β’ ((β:ββΆβ β§ πΉ:dom πΉβΆβ) β (β β πΉ):dom πΉβΆβ) | |
4 | 1, 2, 3 | sylancr 587 | . . 3 β’ (πΉ β MblFn β (β β πΉ):dom πΉβΆβ) |
5 | fimacnv 6736 | . . 3 β’ ((β β πΉ):dom πΉβΆβ β (β‘(β β πΉ) β β) = dom πΉ) | |
6 | 4, 5 | syl 17 | . 2 β’ (πΉ β MblFn β (β‘(β β πΉ) β β) = dom πΉ) |
7 | imaeq2 6053 | . . . 4 β’ (π₯ = β β (β‘(β β πΉ) β π₯) = (β‘(β β πΉ) β β)) | |
8 | 7 | eleq1d 2818 | . . 3 β’ (π₯ = β β ((β‘(β β πΉ) β π₯) β dom vol β (β‘(β β πΉ) β β) β dom vol)) |
9 | ismbf1 25132 | . . . 4 β’ (πΉ β MblFn β (πΉ β (β βpm β) β§ βπ₯ β ran (,)((β‘(β β πΉ) β π₯) β dom vol β§ (β‘(β β πΉ) β π₯) β dom vol))) | |
10 | simpl 483 | . . . . 5 β’ (((β‘(β β πΉ) β π₯) β dom vol β§ (β‘(β β πΉ) β π₯) β dom vol) β (β‘(β β πΉ) β π₯) β dom vol) | |
11 | 10 | ralimi 3083 | . . . 4 β’ (βπ₯ β ran (,)((β‘(β β πΉ) β π₯) β dom vol β§ (β‘(β β πΉ) β π₯) β dom vol) β βπ₯ β ran (,)(β‘(β β πΉ) β π₯) β dom vol) |
12 | 9, 11 | simplbiim 505 | . . 3 β’ (πΉ β MblFn β βπ₯ β ran (,)(β‘(β β πΉ) β π₯) β dom vol) |
13 | ioomax 13395 | . . . . 5 β’ (-β(,)+β) = β | |
14 | ioof 13420 | . . . . . . 7 β’ (,):(β* Γ β*)βΆπ« β | |
15 | ffn 6714 | . . . . . . 7 β’ ((,):(β* Γ β*)βΆπ« β β (,) Fn (β* Γ β*)) | |
16 | 14, 15 | ax-mp 5 | . . . . . 6 β’ (,) Fn (β* Γ β*) |
17 | mnfxr 11267 | . . . . . 6 β’ -β β β* | |
18 | pnfxr 11264 | . . . . . 6 β’ +β β β* | |
19 | fnovrn 7578 | . . . . . 6 β’ (((,) Fn (β* Γ β*) β§ -β β β* β§ +β β β*) β (-β(,)+β) β ran (,)) | |
20 | 16, 17, 18, 19 | mp3an 1461 | . . . . 5 β’ (-β(,)+β) β ran (,) |
21 | 13, 20 | eqeltrri 2830 | . . . 4 β’ β β ran (,) |
22 | 21 | a1i 11 | . . 3 β’ (πΉ β MblFn β β β ran (,)) |
23 | 8, 12, 22 | rspcdva 3613 | . 2 β’ (πΉ β MblFn β (β‘(β β πΉ) β β) β dom vol) |
24 | 6, 23 | eqeltrrd 2834 | 1 β’ (πΉ β MblFn β dom πΉ β dom vol) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 π« cpw 4601 Γ cxp 5673 β‘ccnv 5674 dom cdm 5675 ran crn 5676 β cima 5678 β ccom 5679 Fn wfn 6535 βΆwf 6536 (class class class)co 7405 βpm cpm 8817 βcc 11104 βcr 11105 +βcpnf 11241 -βcmnf 11242 β*cxr 11243 (,)cioo 13320 βcre 15040 βcim 15041 volcvol 24971 MblFncmbf 25122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-er 8699 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-2 12271 df-ioo 13324 df-cj 15042 df-re 15043 df-mbf 25127 |
This theorem is referenced by: ismbf 25136 ismbfcn 25137 mbfimaicc 25139 mbfdm2 25145 mbfres 25152 mbfmulc2lem 25155 mbfimaopn2 25165 cncombf 25166 mbfaddlem 25168 mbfadd 25169 mbfsub 25170 mbfmullem2 25233 mbfmul 25235 bddmulibl 25347 bddibl 25348 bddiblnc 25350 itgulm 25911 ftc1anclem1 36549 ftc1anclem5 36553 ftc1anclem8 36556 smfmbfcex 45462 |
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