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Theorem mbff 22200
Description: A measurable function is a function into the complex numbers. (Contributed by Mario Carneiro, 17-Jun-2014.)
Assertion
Ref Expression
mbff  |-  ( F  e. MblFn  ->  F : dom  F --> CC )

Proof of Theorem mbff
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ismbf1 22199 . . 3  |-  ( 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 )
) )
21simplbi 458 . 2  |-  ( F  e. MblFn  ->  F  e.  ( CC  ^pm  RR )
)
3 cnex 9562 . . . 4  |-  CC  e.  _V
4 reex 9572 . . . 4  |-  RR  e.  _V
53, 4elpm2 7443 . . 3  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
65simplbi 458 . 2  |-  ( F  e.  ( CC  ^pm  RR )  ->  F : dom  F --> CC )
72, 6syl 16 1  |-  ( F  e. MblFn  ->  F : dom  F --> CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   A.wral 2804    C_ wss 3461   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991    o. ccom 4992   -->wf 5566  (class class class)co 6270    ^pm cpm 7413   CCcc 9479   RRcr 9480   (,)cioo 11532   Recre 13012   Imcim 13013   volcvol 22041  MblFncmbf 22189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-pm 7415  df-mbf 22194
This theorem is referenced by:  mbfdm  22201  mbfmptcl  22210  mbfres  22217  mbfimaopnlem  22228  mbfadd  22234  mbfsub  22235  mbfmul  22299  iblcnlem  22361  bddmulibl  22411  bddibl  22412  bddiblnc  30325
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