![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 14998 | . . . 4 β’ β:ββΆβ | |
2 | mbff 24992 | . . . 4 β’ (πΉ β MblFn β πΉ:dom πΉβΆβ) | |
3 | fco 6693 | . . . 4 β’ ((β:ββΆβ β§ πΉ:dom πΉβΆβ) β (β β πΉ):dom πΉβΆβ) | |
4 | 1, 2, 3 | sylancr 588 | . . 3 β’ (πΉ β MblFn β (β β πΉ):dom πΉβΆβ) |
5 | fimacnv 6691 | . . 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 6010 | . . . 4 β’ (π₯ = β β (β‘(β β πΉ) β π₯) = (β‘(β β πΉ) β β)) | |
8 | 7 | eleq1d 2823 | . . 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 24991 | . . . 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 484 | . . . . 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 3087 | . . . 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 506 | . . 3 β’ (πΉ β MblFn β βπ₯ β ran (,)(β‘(β β πΉ) β π₯) β dom vol) |
13 | ioomax 13340 | . . . . 5 β’ (-β(,)+β) = β | |
14 | ioof 13365 | . . . . . . 7 β’ (,):(β* Γ β*)βΆπ« β | |
15 | ffn 6669 | . . . . . . 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 11213 | . . . . . 6 β’ -β β β* | |
18 | pnfxr 11210 | . . . . . 6 β’ +β β β* | |
19 | fnovrn 7530 | . . . . . 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 1462 | . . . . 5 β’ (-β(,)+β) β ran (,) |
21 | 13, 20 | eqeltrri 2835 | . . . 4 β’ β β ran (,) |
22 | 21 | a1i 11 | . . 3 β’ (πΉ β MblFn β β β ran (,)) |
23 | 8, 12, 22 | rspcdva 3583 | . 2 β’ (πΉ β MblFn β (β‘(β β πΉ) β β) β dom vol) |
24 | 6, 23 | eqeltrrd 2839 | 1 β’ (πΉ β MblFn β dom πΉ β dom vol) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 π« cpw 4561 Γ cxp 5632 β‘ccnv 5633 dom cdm 5634 ran crn 5635 β cima 5637 β ccom 5638 Fn wfn 6492 βΆwf 6493 (class class class)co 7358 βpm cpm 8767 βcc 11050 βcr 11051 +βcpnf 11187 -βcmnf 11188 β*cxr 11189 (,)cioo 13265 βcre 14983 βcim 14984 volcvol 24830 MblFncmbf 24981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8649 df-pm 8769 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-2 12217 df-ioo 13269 df-cj 14985 df-re 14986 df-mbf 24986 |
This theorem is referenced by: ismbf 24995 ismbfcn 24996 mbfimaicc 24998 mbfdm2 25004 mbfres 25011 mbfmulc2lem 25014 mbfimaopn2 25024 cncombf 25025 mbfaddlem 25027 mbfadd 25028 mbfsub 25029 mbfmullem2 25092 mbfmul 25094 bddmulibl 25206 bddibl 25207 bddiblnc 25209 itgulm 25770 ftc1anclem1 36154 ftc1anclem5 36158 ftc1anclem8 36161 smfmbfcex 45008 |
Copyright terms: Public domain | W3C validator |