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Theorem measbase 26563
Description: The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
measbase  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )

Proof of Theorem measbase
Dummy variables  x  m  y  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 5711 . 2  |-  ( M  e.  (measures `  S
)  ->  S  e.  dom measures )
2 vex 2970 . . . . 5  |-  s  e. 
_V
3 ovex 6111 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
4 mapex 7212 . . . . 5  |-  ( ( s  e.  _V  /\  ( 0 [,] +oo )  e.  _V )  ->  { m  |  m : s --> ( 0 [,] +oo ) }  e.  _V )
52, 3, 4mp2an 672 . . . 4  |-  { m  |  m : s --> ( 0 [,] +oo ) }  e.  _V
6 simp1 988 . . . . 5  |-  ( ( 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 ) ) )  ->  m : s --> ( 0 [,] +oo ) )
76ss2abi 3419 . . . 4  |-  { 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 ) ) ) }  C_  { m  |  m : s --> ( 0 [,] +oo ) }
85, 7ssexi 4432 . . 3  |-  { 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 ) ) ) }  e.  _V
9 df-meas 26562 . . 3  |- 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 ) ) ) } )
108, 9dmmpti 5535 . 2  |-  dom measures  =  U. ran sigAlgebra
111, 10syl6eleq 2528 1  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {cab 2424   A.wral 2710   _Vcvv 2967   (/)c0 3632   ~Pcpw 3855   U.cuni 4086  Disj wdisj 4257   class class class wbr 4287   dom cdm 4835   ran crn 4836   -->wf 5409   ` cfv 5413  (class class class)co 6086   omcom 6471    ~<_ cdom 7300   0cc0 9274   +oocpnf 9407   [,]cicc 11295  Σ*cesum 26435  sigAlgebracsiga 26502  measurescmeas 26561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6089  df-meas 26562
This theorem is referenced by:  measfrge0  26569  measvnul  26572  measvun  26575  measxun2  26576  measun  26577  measvuni  26580  measssd  26581  measunl  26582  measiuns  26583  measiun  26584  meascnbl  26585  measinblem  26586  measinb  26587  measinb2  26589  measdivcstOLD  26590  measdivcst  26591  aean  26612  mbfmbfm  26625  domprobsiga  26746  prob01  26748  probfinmeasbOLD  26763  probfinmeasb  26764  probmeasb  26765
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