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Theorem ismbf 19001
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 18901. (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 18999 . . 3  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
2 fdm 5409 . . . 4  |-  ( F : A --> RR  ->  dom 
F  =  A )
32eleq1d 2362 . . 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 10740 . . . . 5  |-  (  -oo (,) 
+oo )  =  RR
6 ioorebas 10761 . . . . 5  |-  (  -oo (,) 
+oo )  e.  ran  (,)
75, 6eqeltrri 2367 . . . 4  |-  RR  e.  ran  (,)
8 imaeq2 5024 . . . . . 6  |-  ( x  =  RR  ->  ( `' F " x )  =  ( `' F " RR ) )
98eleq1d 2362 . . . . 5  |-  ( x  =  RR  ->  (
( `' F "
x )  e.  dom  vol  <->  ( `' F " RR )  e.  dom  vol )
)
109rspcv 2893 . . . 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 5673 . . . 4  |-  ( F : A --> RR  ->  ( `' F " RR )  =  A )
1312eleq1d 2362 . . 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 5679 . . . . . . . . . . . . . 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 11725 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  (
Re `  ( F `  x ) )  =  ( F `  x
) )
1817mpteq2dva 4122 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( x  e.  A  |->  ( Re `  ( F `  x )
) )  =  ( x  e.  A  |->  ( F `  x ) ) )
1916recnd 8877 . . . . . . . . . . . 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 5591 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )
22 ref 11613 . . . . . . . . . . . . . 14  |-  Re : CC
--> RR
2322a1i 10 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Re : CC --> RR )
2423feqmptd 5591 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Re  =  ( y  e.  CC  |->  ( Re
`  y ) ) )
25 fveq2 5541 . . . . . . . . . . . 12  |-  ( y  =  ( F `  x )  ->  (
Re `  y )  =  ( Re `  ( F `  x ) ) )
2619, 21, 24, 25fmptco 5707 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Re  o.  F
)  =  ( x  e.  A  |->  ( Re
`  ( F `  x ) ) ) )
2718, 26, 213eqtr4rd 2339 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F  =  ( Re  o.  F ) )
2827cnveqd 4873 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  `' F  =  `' ( Re  o.  F
) )
2928imaeq1d 5027 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' F "
x )  =  ( `' ( Re  o.  F ) " x
) )
3029eleq1d 2362 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( ( `' F " x )  e.  dom  vol  <->  ( `' ( Re  o.  F ) " x
)  e.  dom  vol ) )
31 imf 11614 . . . . . . . . . . . . . . . 16  |-  Im : CC
--> RR
3231a1i 10 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Im : CC --> RR )
3332feqmptd 5591 . . . . . . . . . . . . . 14  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Im  =  ( y  e.  CC  |->  ( Im
`  y ) ) )
34 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( y  =  ( F `  x )  ->  (
Im `  y )  =  ( Im `  ( F `  x ) ) )
3519, 21, 33, 34fmptco 5707 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( x  e.  A  |->  ( Im
`  ( F `  x ) ) ) )
3616reim0d 11726 . . . . . . . . . . . . . 14  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  (
Im `  ( F `  x ) )  =  0 )
3736mpteq2dva 4122 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( x  e.  A  |->  ( Im `  ( F `  x )
) )  =  ( x  e.  A  |->  0 ) )
3835, 37eqtrd 2328 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( x  e.  A  |->  0 ) )
39 fconstmpt 4748 . . . . . . . . . . . 12  |-  ( A  X.  { 0 } )  =  ( x  e.  A  |->  0 )
4038, 39syl6eqr 2346 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( A  X.  { 0 } ) )
4140cnveqd 4873 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  `' ( Im  o.  F )  =  `' ( A  X.  { 0 } ) )
4241imaeq1d 5027 . . . . . . . . 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 8854 . . . . . . . . . 10  |-  0  e.  RR
45 mbfconstlem 19000 . . . . . . . . . 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 2370 . . . . . . . 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 2576 . . . . 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 8810 . . . . . . . 8  |-  RR  C_  CC
52 fss 5413 . . . . . . . 8  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
5351, 52mpan2 652 . . . . . . 7  |-  ( F : A --> RR  ->  F : A --> CC )
54 mblss 18906 . . . . . . 7  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
55 cnex 8834 . . . . . . . 8  |-  CC  e.  _V
56 reex 8844 . . . . . . . 8  |-  RR  e.  _V
57 elpm2r 6804 . . . . . . . 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 18997 . . . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   {csn 3653    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^pm cpm 6789   CCcc 8751   RRcr 8752   0cc0 8753    +oocpnf 8880    -oocmnf 8881   (,)cioo 10672   Recre 11598   Imcim 11599   volcvol 18839  MblFncmbf 18985
This theorem is referenced by:  ismbfcn  19002  mbfima  19003  mbfid  19007  ismbfd  19011  mbfeqalem  19013  mbfres2  19016  mbfimaopnlem  19026  i1fd  19052  elmbfmvol2  23587
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xadd 10469  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-xmet 16389  df-met 16390  df-ovol 18840  df-vol 18841  df-mbf 18991
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