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Theorem ismbf 18985
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 18885. (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 18983 . . 3  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
2 fdm 5393 . . . 4  |-  ( F : A --> RR  ->  dom 
F  =  A )
32eleq1d 2349 . . 3  |-  ( F : A --> RR  ->  ( dom  F  e.  dom  vol  <->  A  e.  dom  vol )
)
41, 3syl5ib 210 . 2  |-  ( F : A --> RR  ->  ( F  e. MblFn  ->  A  e. 
dom  vol ) )
5 ioomax 10724 . . . . 5  |-  (  -oo (,) 
+oo )  =  RR
6 ioorebas 10745 . . . . 5  |-  (  -oo (,) 
+oo )  e.  ran  (,)
75, 6eqeltrri 2354 . . . 4  |-  RR  e.  ran  (,)
8 imaeq2 5008 . . . . . 6  |-  ( x  =  RR  ->  ( `' F " x )  =  ( `' F " RR ) )
98eleq1d 2349 . . . . 5  |-  ( x  =  RR  ->  (
( `' F "
x )  e.  dom  vol  <->  ( `' F " RR )  e.  dom  vol )
)
109rspcv 2880 . . . 4  |-  ( RR  e.  ran  (,)  ->  ( A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol 
->  ( `' F " RR )  e.  dom  vol ) )
117, 10ax-mp 8 . . 3  |-  ( A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol  ->  ( `' F " RR )  e.  dom  vol )
12 fimacnv 5657 . . . 4  |-  ( F : A --> RR  ->  ( `' F " RR )  =  A )
1312eleq1d 2349 . . 3  |-  ( F : A --> RR  ->  ( ( `' F " RR )  e.  dom  vol  <->  A  e.  dom  vol )
)
1411, 13syl5ib 210 . 2  |-  ( F : A --> RR  ->  ( A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol 
->  A  e.  dom  vol ) )
15 ffvelrn 5663 . . . . . . . . . . . . . 14  |-  ( ( F : A --> RR  /\  x  e.  A )  ->  ( F `  x
)  e.  RR )
1615adantlr 695 . . . . . . . . . . . . 13  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  ( F `  x )  e.  RR )
1716rered 11709 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  (
Re `  ( F `  x ) )  =  ( F `  x
) )
1817mpteq2dva 4106 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( x  e.  A  |->  ( Re `  ( F `  x )
) )  =  ( x  e.  A  |->  ( F `  x ) ) )
1916recnd 8861 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  ( F `  x )  e.  CC )
20 simpl 443 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F : A --> RR )
2120feqmptd 5575 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )
22 ref 11597 . . . . . . . . . . . . . 14  |-  Re : CC
--> RR
2322a1i 10 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Re : CC --> RR )
2423feqmptd 5575 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Re  =  ( y  e.  CC  |->  ( Re
`  y ) ) )
25 fveq2 5525 . . . . . . . . . . . 12  |-  ( y  =  ( F `  x )  ->  (
Re `  y )  =  ( Re `  ( F `  x ) ) )
2619, 21, 24, 25fmptco 5691 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Re  o.  F
)  =  ( x  e.  A  |->  ( Re
`  ( F `  x ) ) ) )
2718, 26, 213eqtr4rd 2326 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F  =  ( Re  o.  F ) )
2827cnveqd 4857 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  `' F  =  `' ( Re  o.  F
) )
2928imaeq1d 5011 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' F "
x )  =  ( `' ( Re  o.  F ) " x
) )
3029eleq1d 2349 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( ( `' F " x )  e.  dom  vol  <->  ( `' ( Re  o.  F ) " x
)  e.  dom  vol ) )
31 imf 11598 . . . . . . . . . . . . . . . 16  |-  Im : CC
--> RR
3231a1i 10 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Im : CC --> RR )
3332feqmptd 5575 . . . . . . . . . . . . . 14  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Im  =  ( y  e.  CC  |->  ( Im
`  y ) ) )
34 fveq2 5525 . . . . . . . . . . . . . 14  |-  ( y  =  ( F `  x )  ->  (
Im `  y )  =  ( Im `  ( F `  x ) ) )
3519, 21, 33, 34fmptco 5691 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( x  e.  A  |->  ( Im
`  ( F `  x ) ) ) )
3616reim0d 11710 . . . . . . . . . . . . . 14  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  (
Im `  ( F `  x ) )  =  0 )
3736mpteq2dva 4106 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( x  e.  A  |->  ( Im `  ( F `  x )
) )  =  ( x  e.  A  |->  0 ) )
3835, 37eqtrd 2315 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( x  e.  A  |->  0 ) )
39 fconstmpt 4732 . . . . . . . . . . . 12  |-  ( A  X.  { 0 } )  =  ( x  e.  A  |->  0 )
4038, 39syl6eqr 2333 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( A  X.  { 0 } ) )
4140cnveqd 4857 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  `' ( Im  o.  F )  =  `' ( A  X.  { 0 } ) )
4241imaeq1d 5011 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' ( Im  o.  F ) "
x )  =  ( `' ( A  X.  { 0 } )
" x ) )
43 simpr 447 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  A  e.  dom  vol )
44 0re 8838 . . . . . . . . . 10  |-  0  e.  RR
45 mbfconstlem 18984 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  0  e.  RR )  ->  ( `' ( A  X.  { 0 } ) " x
)  e.  dom  vol )
4643, 44, 45sylancl 643 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' ( A  X.  { 0 } ) " x )  e.  dom  vol )
4742, 46eqeltrd 2357 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' ( Im  o.  F ) "
x )  e.  dom  vol )
4847biantrud 493 . . . . . . 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 244 . . . . . 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 2563 . . . . 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 8794 . . . . . . . 8  |-  RR  C_  CC
52 fss 5397 . . . . . . . 8  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
5351, 52mpan2 652 . . . . . . 7  |-  ( F : A --> RR  ->  F : A --> CC )
54 mblss 18890 . . . . . . 7  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
55 cnex 8818 . . . . . . . 8  |-  CC  e.  _V
56 reex 8828 . . . . . . . 8  |-  RR  e.  _V
57 elpm2r 6788 . . . . . . . 8  |-  ( ( ( CC  e.  _V  /\  RR  e.  _V )  /\  ( F : A --> CC  /\  A  C_  RR ) )  ->  F  e.  ( CC  ^pm  RR ) )
5855, 56, 57mpanl12 663 . . . . . . 7  |-  ( ( F : A --> CC  /\  A  C_  RR )  ->  F  e.  ( CC  ^pm 
RR ) )
5953, 54, 58syl2an 463 . . . . . 6  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F  e.  ( CC 
^pm  RR ) )
6059biantrurd 494 . . . . 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 244 . . . 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 18981 . . . 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 255 . . 3  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
)
6463ex 423 . 2  |-  ( F : A --> RR  ->  ( A  e.  dom  vol  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
) )
654, 14, 64pm5.21ndd 343 1  |-  ( F : A --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^pm cpm 6773   CCcc 8735   RRcr 8736   0cc0 8737    +oocpnf 8864    -oocmnf 8865   (,)cioo 10656   Recre 11582   Imcim 11583   volcvol 18823  MblFncmbf 18969
This theorem is referenced by:  ismbfcn  18986  mbfima  18987  mbfid  18991  ismbfd  18995  mbfeqalem  18997  mbfres2  19000  mbfimaopnlem  19010  i1fd  19036  elmbfmvol2  23572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824  df-vol 18825  df-mbf 18975
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