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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 15061 | . . . 4 β’ β:ββΆβ | |
2 | mbff 25498 | . . . 4 β’ (πΉ β MblFn β πΉ:dom πΉβΆβ) | |
3 | fco 6732 | . . . 4 β’ ((β:ββΆβ β§ πΉ:dom πΉβΆβ) β (β β πΉ):dom πΉβΆβ) | |
4 | 1, 2, 3 | sylancr 586 | . . 3 β’ (πΉ β MblFn β (β β πΉ):dom πΉβΆβ) |
5 | fimacnv 6730 | . . 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 6046 | . . . 4 β’ (π₯ = β β (β‘(β β πΉ) β π₯) = (β‘(β β πΉ) β β)) | |
8 | 7 | eleq1d 2810 | . . 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 25497 | . . . 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 482 | . . . . 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 3075 | . . . 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 504 | . . 3 β’ (πΉ β MblFn β βπ₯ β ran (,)(β‘(β β πΉ) β π₯) β dom vol) |
13 | ioomax 13400 | . . . . 5 β’ (-β(,)+β) = β | |
14 | ioof 13425 | . . . . . . 7 β’ (,):(β* Γ β*)βΆπ« β | |
15 | ffn 6708 | . . . . . . 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 11270 | . . . . . 6 β’ -β β β* | |
18 | pnfxr 11267 | . . . . . 6 β’ +β β β* | |
19 | fnovrn 7576 | . . . . . 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 1457 | . . . . 5 β’ (-β(,)+β) β ran (,) |
21 | 13, 20 | eqeltrri 2822 | . . . 4 β’ β β ran (,) |
22 | 21 | a1i 11 | . . 3 β’ (πΉ β MblFn β β β ran (,)) |
23 | 8, 12, 22 | rspcdva 3605 | . 2 β’ (πΉ β MblFn β (β‘(β β πΉ) β β) β dom vol) |
24 | 6, 23 | eqeltrrd 2826 | 1 β’ (πΉ β MblFn β dom πΉ β dom vol) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3053 π« cpw 4595 Γ cxp 5665 β‘ccnv 5666 dom cdm 5667 ran crn 5668 β cima 5670 β ccom 5671 Fn wfn 6529 βΆwf 6530 (class class class)co 7402 βpm cpm 8818 βcc 11105 βcr 11106 +βcpnf 11244 -βcmnf 11245 β*cxr 11246 (,)cioo 13325 βcre 15046 βcim 15047 volcvol 25336 MblFncmbf 25487 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-er 8700 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-2 12274 df-ioo 13329 df-cj 15048 df-re 15049 df-mbf 25492 |
This theorem is referenced by: ismbf 25501 ismbfcn 25502 mbfimaicc 25504 mbfdm2 25510 mbfres 25517 mbfmulc2lem 25520 mbfimaopn2 25530 cncombf 25531 mbfaddlem 25533 mbfadd 25534 mbfsub 25535 mbfmullem2 25598 mbfmul 25600 bddmulibl 25712 bddibl 25713 bddiblnc 25715 itgulm 26285 ftc1anclem1 37065 ftc1anclem5 37069 ftc1anclem8 37072 smfmbfcex 46022 |
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