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Theorem measbase 29019
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 5891 . 2  |-  ( M  e.  (measures `  S
)  ->  S  e.  dom measures )
2 vex 3048 . . . . 5  |-  s  e. 
_V
3 ovex 6318 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
4 mapex 7478 . . . . 5  |-  ( ( s  e.  _V  /\  ( 0 [,] +oo )  e.  _V )  ->  { m  |  m : s --> ( 0 [,] +oo ) }  e.  _V )
52, 3, 4mp2an 678 . . . 4  |-  { m  |  m : s --> ( 0 [,] +oo ) }  e.  _V
6 simp1 1008 . . . . 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 3501 . . . 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 4548 . . 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 29018 . . 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 5707 . 2  |-  dom measures  =  U. ran sigAlgebra
111, 10syl6eleq 2539 1  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   {cab 2437   A.wral 2737   _Vcvv 3045   (/)c0 3731   ~Pcpw 3951   U.cuni 4198  Disj wdisj 4373   class class class wbr 4402   dom cdm 4834   ran crn 4835   -->wf 5578   ` cfv 5582  (class class class)co 6290   omcom 6692    ~<_ cdom 7567   0cc0 9539   +oocpnf 9672   [,]cicc 11638  Σ*cesum 28848  sigAlgebracsiga 28929  measurescmeas 29017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590  df-ov 6293  df-meas 29018
This theorem is referenced by:  measfrge0  29025  measvnul  29028  measvun  29031  measxun2  29032  measun  29033  measvuni  29036  measssd  29037  measunl  29038  measiuns  29039  measiun  29040  meascnbl  29041  measinblem  29042  measinb  29043  measinb2  29045  measdivcstOLD  29046  measdivcst  29047  aean  29067  mbfmbfm  29080  domprobsiga  29244  prob01  29246  probfinmeasbOLD  29261  probfinmeasb  29262  probmeasb  29263
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