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Theorem measbase 28631
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 5874 . 2  |-  ( M  e.  (measures `  S
)  ->  S  e.  dom measures )
2 vex 3061 . . . . 5  |-  s  e. 
_V
3 ovex 6305 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
4 mapex 7462 . . . . 5  |-  ( ( s  e.  _V  /\  ( 0 [,] +oo )  e.  _V )  ->  { m  |  m : s --> ( 0 [,] +oo ) }  e.  _V )
52, 3, 4mp2an 670 . . . 4  |-  { m  |  m : s --> ( 0 [,] +oo ) }  e.  _V
6 simp1 997 . . . . 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 3510 . . . 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 4538 . . 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 28630 . . 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 5692 . 2  |-  dom measures  =  U. ran sigAlgebra
111, 10syl6eleq 2500 1  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   {cab 2387   A.wral 2753   _Vcvv 3058   (/)c0 3737   ~Pcpw 3954   U.cuni 4190  Disj wdisj 4365   class class class wbr 4394   dom cdm 4822   ran crn 4823   -->wf 5564   ` cfv 5568  (class class class)co 6277   omcom 6682    ~<_ cdom 7551   0cc0 9521   +oocpnf 9654   [,]cicc 11584  Σ*cesum 28460  sigAlgebracsiga 28541  measurescmeas 28629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-ov 6280  df-meas 28630
This theorem is referenced by:  measfrge0  28637  measvnul  28640  measvun  28643  measxun2  28644  measun  28645  measvuni  28648  measssd  28649  measunl  28650  measiuns  28651  measiun  28652  meascnbl  28653  measinblem  28654  measinb  28655  measinb2  28657  measdivcstOLD  28658  measdivcst  28659  aean  28679  mbfmbfm  28692  domprobsiga  28842  prob01  28844  probfinmeasbOLD  28859  probfinmeasb  28860  probmeasb  28861
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