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Theorem ismbf1 19508
Description: The predicate " F is a measurable function". This is more naturally stated for functions on the reals, see ismbf 19512 and ismbfcn 19513 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 5023 . . . . . . 7  |-  ( f  =  F  ->  (
Re  o.  f )  =  ( Re  o.  F ) )
21cnveqd 5040 . . . . . 6  |-  ( f  =  F  ->  `' ( Re  o.  f
)  =  `' ( Re  o.  F ) )
32imaeq1d 5194 . . . . 5  |-  ( f  =  F  ->  ( `' ( Re  o.  f ) " x
)  =  ( `' ( Re  o.  F
) " x ) )
43eleq1d 2501 . . . 4  |-  ( f  =  F  ->  (
( `' ( Re  o.  f ) "
x )  e.  dom  vol  <->  ( `' ( Re  o.  F ) " x
)  e.  dom  vol ) )
5 coeq2 5023 . . . . . . 7  |-  ( f  =  F  ->  (
Im  o.  f )  =  ( Im  o.  F ) )
65cnveqd 5040 . . . . . 6  |-  ( f  =  F  ->  `' ( Im  o.  f
)  =  `' ( Im  o.  F ) )
76imaeq1d 5194 . . . . 5  |-  ( f  =  F  ->  ( `' ( Im  o.  f ) " x
)  =  ( `' ( Im  o.  F
) " x ) )
87eleq1d 2501 . . . 4  |-  ( f  =  F  ->  (
( `' ( Im  o.  f ) "
x )  e.  dom  vol  <->  ( `' ( Im  o.  F ) " x
)  e.  dom  vol ) )
94, 8anbi12d 692 . . 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 2717 . 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 19502 . 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 3086 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   `'ccnv 4869   dom cdm 4870   ran crn 4871   "cima 4873    o. ccom 4874  (class class class)co 6073    ^pm cpm 7011   CCcc 8978   RRcr 8979   (,)cioo 10906   Recre 11892   Imcim 11893   volcvol 19350  MblFncmbf 19496
This theorem is referenced by:  mbff  19509  mbfdm  19510  ismbf  19512  ismbfcn  19513  mbfconst  19517  mbfres  19526  cncombf  19540  cnmbf  19541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-mbf 19502
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