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Theorem measbasedom 26752
Description: The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
measbasedom  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )

Proof of Theorem measbasedom
Dummy variables  x  y  m  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isrnmeas 26750 . . . 4  |-  ( M  e.  U. ran measures  ->  ( dom  M  e.  U. ran sigAlgebra  /\  ( M : dom  M --> ( 0 [,] +oo )  /\  ( M `  (/) )  =  0  /\ 
A. x  e.  ~P  dom  M ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  ( M `  U. x )  = Σ* y  e.  x ( M `  y ) ) ) ) )
21simprd 463 . . 3  |-  ( M  e.  U. ran measures  ->  ( M : dom  M --> ( 0 [,] +oo )  /\  ( M `  (/) )  =  0  /\  A. x  e.  ~P  dom  M ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  ( M `  U. x )  = Σ* y  e.  x ( M `  y ) ) ) )
31simpld 459 . . . 4  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
4 ismeas 26749 . . . 4  |-  ( dom 
M  e.  U. ran sigAlgebra  ->  ( M  e.  (measures `  dom  M )  <->  ( M : dom  M --> ( 0 [,] +oo )  /\  ( M `  (/) )  =  0  /\  A. x  e.  ~P  dom  M ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  ( M `  U. x )  = Σ* y  e.  x ( M `  y ) ) ) ) )
53, 4syl 16 . . 3  |-  ( M  e.  U. ran measures  ->  ( M  e.  (measures `  dom  M )  <->  ( M : dom  M --> ( 0 [,] +oo )  /\  ( M `  (/) )  =  0  /\  A. x  e.  ~P  dom  M ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  ( M `  U. x )  = Σ* y  e.  x ( M `  y ) ) ) ) )
62, 5mpbird 232 . 2  |-  ( M  e.  U. ran measures  ->  M  e.  (measures `  dom  M ) )
7 df-meas 26746 . . . 4  |- measures  =  ( s  e.  U. ran sigAlgebra  |->  { m  |  ( m : s --> ( 0 [,] +oo )  /\  ( m `  (/) )  =  0  /\  A. x  e.  ~P  s ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) } )
87funmpt2 5555 . . 3  |-  Fun measures
9 elunirn2 26102 . . 3  |-  ( ( Fun measures  /\  M  e.  (measures `  dom  M ) )  ->  M  e.  U. ran measures )
108, 9mpan 670 . 2  |-  ( M  e.  (measures `  dom  M )  ->  M  e.  U.
ran measures )
116, 10impbii 188 1  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {cab 2436   A.wral 2795   (/)c0 3737   ~Pcpw 3960   U.cuni 4191  Disj wdisj 4362   class class class wbr 4392   dom cdm 4940   ran crn 4941   Fun wfun 5512   -->wf 5514   ` cfv 5518  (class class class)co 6192   omcom 6578    ~<_ cdom 7410   0cc0 9385   +oocpnf 9518   [,]cicc 11406  Σ*cesum 26619  sigAlgebracsiga 26686  measurescmeas 26745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-ov 6195  df-esum 26620  df-meas 26746
This theorem is referenced by:  truae  26795  aean  26796  mbfmbfm  26809  sibfinima  26861  sibfof  26862  domprobmeas  26929  probmeasd  26942  probfinmeasbOLD  26947  probfinmeasb  26948  probmeasb  26949  dstrvprob  26990
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