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Theorem ismbf1 18981
Description: The predicate " F is a measurable function". This is more naturally stated for functions on the reals, see ismbf 18985 and ismbfcn 18986 for the decomposition of the real and imaginary parts. (Contributed by Mario Carneiro, 17-Jun-2014.)
Assertion
Ref Expression
ismbf1  |-  ( 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 )
) )
Distinct variable group:    x, F

Proof of Theorem ismbf1
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 coeq2 4842 . . . . . . 7  |-  ( f  =  F  ->  (
Re  o.  f )  =  ( Re  o.  F ) )
21cnveqd 4857 . . . . . 6  |-  ( f  =  F  ->  `' ( Re  o.  f
)  =  `' ( Re  o.  F ) )
32imaeq1d 5011 . . . . 5  |-  ( f  =  F  ->  ( `' ( Re  o.  f ) " x
)  =  ( `' ( Re  o.  F
) " x ) )
43eleq1d 2349 . . . 4  |-  ( f  =  F  ->  (
( `' ( Re  o.  f ) "
x )  e.  dom  vol  <->  ( `' ( Re  o.  F ) " x
)  e.  dom  vol ) )
5 coeq2 4842 . . . . . . 7  |-  ( f  =  F  ->  (
Im  o.  f )  =  ( Im  o.  F ) )
65cnveqd 4857 . . . . . 6  |-  ( f  =  F  ->  `' ( Im  o.  f
)  =  `' ( Im  o.  F ) )
76imaeq1d 5011 . . . . 5  |-  ( f  =  F  ->  ( `' ( Im  o.  f ) " x
)  =  ( `' ( Im  o.  F
) " x ) )
87eleq1d 2349 . . . 4  |-  ( f  =  F  ->  (
( `' ( Im  o.  f ) "
x )  e.  dom  vol  <->  ( `' ( Im  o.  F ) " x
)  e.  dom  vol ) )
94, 8anbi12d 691 . . 3  |-  ( f  =  F  ->  (
( ( `' ( Re  o.  f )
" x )  e. 
dom  vol  /\  ( `' ( Im  o.  f
) " x )  e.  dom  vol )  <->  ( ( `' ( Re  o.  F ) "
x )  e.  dom  vol 
/\  ( `' ( Im  o.  F )
" x )  e. 
dom  vol ) ) )
109ralbidv 2563 . 2  |-  ( f  =  F  ->  ( A. x  e.  ran  (,) ( ( `' ( Re  o.  f )
" x )  e. 
dom  vol  /\  ( `' ( Im  o.  f
) " x )  e.  dom  vol )  <->  A. x  e.  ran  (,) ( ( `' ( Re  o.  F )
" x )  e. 
dom  vol  /\  ( `' ( Im  o.  F
) " x )  e.  dom  vol )
) )
11 df-mbf 18975 . 2  |- MblFn  =  {
f  e.  ( CC 
^pm  RR )  |  A. x  e.  ran  (,) (
( `' ( Re  o.  f ) "
x )  e.  dom  vol 
/\  ( `' ( Im  o.  f )
" x )  e. 
dom  vol ) }
1210, 11elrab2 2925 1  |-  ( 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 )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    o. ccom 4693  (class class class)co 5858    ^pm cpm 6773   CCcc 8735   RRcr 8736   (,)cioo 10656   Recre 11582   Imcim 11583   volcvol 18823  MblFncmbf 18969
This theorem is referenced by:  mbff  18982  mbfdm  18983  ismbf  18985  ismbfcn  18986  mbfconst  18990  mbfres  18999  cncombf  19013  cnmbf  19014
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-mbf 18975
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