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Theorem ismbf1 18997
Description: The predicate " F is a measurable function". This is more naturally stated for functions on the reals, see ismbf 19001 and ismbfcn 19002 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 4858 . . . . . . 7  |-  ( f  =  F  ->  (
Re  o.  f )  =  ( Re  o.  F ) )
21cnveqd 4873 . . . . . 6  |-  ( f  =  F  ->  `' ( Re  o.  f
)  =  `' ( Re  o.  F ) )
32imaeq1d 5027 . . . . 5  |-  ( f  =  F  ->  ( `' ( Re  o.  f ) " x
)  =  ( `' ( Re  o.  F
) " x ) )
43eleq1d 2362 . . . 4  |-  ( f  =  F  ->  (
( `' ( Re  o.  f ) "
x )  e.  dom  vol  <->  ( `' ( Re  o.  F ) " x
)  e.  dom  vol ) )
5 coeq2 4858 . . . . . . 7  |-  ( f  =  F  ->  (
Im  o.  f )  =  ( Im  o.  F ) )
65cnveqd 4873 . . . . . 6  |-  ( f  =  F  ->  `' ( Im  o.  f
)  =  `' ( Im  o.  F ) )
76imaeq1d 5027 . . . . 5  |-  ( f  =  F  ->  ( `' ( Im  o.  f ) " x
)  =  ( `' ( Im  o.  F
) " x ) )
87eleq1d 2362 . . . 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 2576 . 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 18991 . 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 2938 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 1632    e. wcel 1696   A.wral 2556   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708    o. ccom 4709  (class class class)co 5874    ^pm cpm 6789   CCcc 8751   RRcr 8752   (,)cioo 10672   Recre 11598   Imcim 11599   volcvol 18839  MblFncmbf 18985
This theorem is referenced by:  mbff  18998  mbfdm  18999  ismbf  19001  ismbfcn  19002  mbfconst  19006  mbfres  19015  cncombf  19029  cnmbf  19030
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-mbf 18991
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