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Theorem ismbf 22463
Description: The predicate " F is a measurable function". A function is measurable iff the preimages of all open intervals are measurable sets in the sense of ismbl 22357. (Contributed by Mario Carneiro, 17-Jun-2014.)
Assertion
Ref Expression
ismbf  |-  ( F : A --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol ) )
Distinct variable groups:    x, F    x, A

Proof of Theorem ismbf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 mbfdm 22461 . . 3  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
2 fdm 5750 . . . 4  |-  ( F : A --> RR  ->  dom 
F  =  A )
32eleq1d 2498 . . 3  |-  ( F : A --> RR  ->  ( dom  F  e.  dom  vol  <->  A  e.  dom  vol )
)
41, 3syl5ib 222 . 2  |-  ( F : A --> RR  ->  ( F  e. MblFn  ->  A  e. 
dom  vol ) )
5 ioomax 11709 . . . . 5  |-  ( -oo (,) +oo )  =  RR
6 ioorebas 11736 . . . . 5  |-  ( -oo (,) +oo )  e.  ran  (,)
75, 6eqeltrri 2514 . . . 4  |-  RR  e.  ran  (,)
8 imaeq2 5184 . . . . . 6  |-  ( x  =  RR  ->  ( `' F " x )  =  ( `' F " RR ) )
98eleq1d 2498 . . . . 5  |-  ( x  =  RR  ->  (
( `' F "
x )  e.  dom  vol  <->  ( `' F " RR )  e.  dom  vol )
)
109rspcv 3184 . . . 4  |-  ( RR  e.  ran  (,)  ->  ( A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol 
->  ( `' F " RR )  e.  dom  vol ) )
117, 10ax-mp 5 . . 3  |-  ( A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol  ->  ( `' F " RR )  e.  dom  vol )
12 fimacnv 6027 . . . 4  |-  ( F : A --> RR  ->  ( `' F " RR )  =  A )
1312eleq1d 2498 . . 3  |-  ( F : A --> RR  ->  ( ( `' F " RR )  e.  dom  vol  <->  A  e.  dom  vol )
)
1411, 13syl5ib 222 . 2  |-  ( F : A --> RR  ->  ( A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol 
->  A  e.  dom  vol ) )
15 ffvelrn 6035 . . . . . . . . . . . . . 14  |-  ( ( F : A --> RR  /\  x  e.  A )  ->  ( F `  x
)  e.  RR )
1615adantlr 719 . . . . . . . . . . . . 13  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  ( F `  x )  e.  RR )
1716rered 13266 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  (
Re `  ( F `  x ) )  =  ( F `  x
) )
1817mpteq2dva 4512 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( x  e.  A  |->  ( Re `  ( F `  x )
) )  =  ( x  e.  A  |->  ( F `  x ) ) )
1916recnd 9668 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  ( F `  x )  e.  CC )
20 simpl 458 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F : A --> RR )
2120feqmptd 5934 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )
22 ref 13154 . . . . . . . . . . . . . 14  |-  Re : CC
--> RR
2322a1i 11 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Re : CC --> RR )
2423feqmptd 5934 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Re  =  ( y  e.  CC  |->  ( Re
`  y ) ) )
25 fveq2 5881 . . . . . . . . . . . 12  |-  ( y  =  ( F `  x )  ->  (
Re `  y )  =  ( Re `  ( F `  x ) ) )
2619, 21, 24, 25fmptco 6071 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Re  o.  F
)  =  ( x  e.  A  |->  ( Re
`  ( F `  x ) ) ) )
2718, 26, 213eqtr4rd 2481 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F  =  ( Re  o.  F ) )
2827cnveqd 5030 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  `' F  =  `' ( Re  o.  F
) )
2928imaeq1d 5187 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' F "
x )  =  ( `' ( Re  o.  F ) " x
) )
3029eleq1d 2498 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( ( `' F " x )  e.  dom  vol  <->  ( `' ( Re  o.  F ) " x
)  e.  dom  vol ) )
31 imf 13155 . . . . . . . . . . . . . . . 16  |-  Im : CC
--> RR
3231a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Im : CC --> RR )
3332feqmptd 5934 . . . . . . . . . . . . . 14  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Im  =  ( y  e.  CC  |->  ( Im
`  y ) ) )
34 fveq2 5881 . . . . . . . . . . . . . 14  |-  ( y  =  ( F `  x )  ->  (
Im `  y )  =  ( Im `  ( F `  x ) ) )
3519, 21, 33, 34fmptco 6071 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( x  e.  A  |->  ( Im
`  ( F `  x ) ) ) )
3616reim0d 13267 . . . . . . . . . . . . . 14  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  (
Im `  ( F `  x ) )  =  0 )
3736mpteq2dva 4512 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( x  e.  A  |->  ( Im `  ( F `  x )
) )  =  ( x  e.  A  |->  0 ) )
3835, 37eqtrd 2470 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( x  e.  A  |->  0 ) )
39 fconstmpt 4898 . . . . . . . . . . . 12  |-  ( A  X.  { 0 } )  =  ( x  e.  A  |->  0 )
4038, 39syl6eqr 2488 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( A  X.  { 0 } ) )
4140cnveqd 5030 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  `' ( Im  o.  F )  =  `' ( A  X.  { 0 } ) )
4241imaeq1d 5187 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' ( Im  o.  F ) "
x )  =  ( `' ( A  X.  { 0 } )
" x ) )
43 simpr 462 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  A  e.  dom  vol )
44 0re 9642 . . . . . . . . . 10  |-  0  e.  RR
45 mbfconstlem 22462 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  0  e.  RR )  ->  ( `' ( A  X.  { 0 } ) " x
)  e.  dom  vol )
4643, 44, 45sylancl 666 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' ( A  X.  { 0 } ) " x )  e.  dom  vol )
4742, 46eqeltrd 2517 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' ( Im  o.  F ) "
x )  e.  dom  vol )
4847biantrud 509 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( ( `' ( Re  o.  F )
" x )  e. 
dom  vol  <->  ( ( `' ( Re  o.  F
) " x )  e.  dom  vol  /\  ( `' ( Im  o.  F ) " x
)  e.  dom  vol ) ) )
4930, 48bitrd 256 . . . . . 6  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( ( `' F " x )  e.  dom  vol  <->  ( ( `' ( Re  o.  F ) "
x )  e.  dom  vol 
/\  ( `' ( Im  o.  F )
" x )  e. 
dom  vol ) ) )
5049ralbidv 2871 . . . . 5  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( A. x  e. 
ran  (,) ( `' F " x )  e.  dom  vol  <->  A. x  e.  ran  (,) ( ( `' ( Re  o.  F )
" x )  e. 
dom  vol  /\  ( `' ( Im  o.  F
) " x )  e.  dom  vol )
) )
51 ax-resscn 9595 . . . . . . . 8  |-  RR  C_  CC
52 fss 5754 . . . . . . . 8  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
5351, 52mpan2 675 . . . . . . 7  |-  ( F : A --> RR  ->  F : A --> CC )
54 mblss 22362 . . . . . . 7  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
55 cnex 9619 . . . . . . . 8  |-  CC  e.  _V
56 reex 9629 . . . . . . . 8  |-  RR  e.  _V
57 elpm2r 7497 . . . . . . . 8  |-  ( ( ( CC  e.  _V  /\  RR  e.  _V )  /\  ( F : A --> CC  /\  A  C_  RR ) )  ->  F  e.  ( CC  ^pm  RR ) )
5855, 56, 57mpanl12 686 . . . . . . 7  |-  ( ( F : A --> CC  /\  A  C_  RR )  ->  F  e.  ( CC  ^pm 
RR ) )
5953, 54, 58syl2an 479 . . . . . 6  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F  e.  ( CC 
^pm  RR ) )
6059biantrurd 510 . . . . 5  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( A. x  e. 
ran  (,) ( ( `' ( Re  o.  F
) " x )  e.  dom  vol  /\  ( `' ( Im  o.  F ) " x
)  e.  dom  vol ) 
<->  ( F  e.  ( CC  ^pm  RR )  /\  A. x  e.  ran  (,) ( ( `' ( Re  o.  F )
" x )  e. 
dom  vol  /\  ( `' ( Im  o.  F
) " x )  e.  dom  vol )
) ) )
6150, 60bitrd 256 . . . 4  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( A. x  e. 
ran  (,) ( `' F " x )  e.  dom  vol  <->  ( F  e.  ( CC 
^pm  RR )  /\  A. x  e.  ran  (,) (
( `' ( Re  o.  F ) "
x )  e.  dom  vol 
/\  ( `' ( Im  o.  F )
" x )  e. 
dom  vol ) ) ) )
62 ismbf1 22459 . . . 4  |-  ( F  e. MblFn 
<->  ( F  e.  ( CC  ^pm  RR )  /\  A. x  e.  ran  (,) ( ( `' ( Re  o.  F )
" x )  e. 
dom  vol  /\  ( `' ( Im  o.  F
) " x )  e.  dom  vol )
) )
6361, 62syl6rbbr 267 . . 3  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
)
6463ex 435 . 2  |-  ( F : A --> RR  ->  ( A  e.  dom  vol  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
) )
654, 14, 64pm5.21ndd 355 1  |-  ( F : A --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087    C_ wss 3442   {csn 4002    |-> cmpt 4484    X. cxp 4852   `'ccnv 4853   dom cdm 4854   ran crn 4855   "cima 4857    o. ccom 4858   -->wf 5597   ` cfv 5601  (class class class)co 6305    ^pm cpm 7481   CCcc 9536   RRcr 9537   0cc0 9538   +oocpnf 9671   -oocmnf 9672   (,)cioo 11635   Recre 13139   Imcim 13140   volcvol 22295  MblFncmbf 22449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xadd 11410  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-xmet 18898  df-met 18899  df-ovol 22296  df-vol 22297  df-mbf 22454
This theorem is referenced by:  ismbfcn  22464  mbfima  22465  mbfid  22469  ismbfd  22473  mbfeqalem  22475  mbfres2  22478  mbfimaopnlem  22488  i1fd  22516  elmbfmvol2  28928  cnambfre  31693  mbf0  37403
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