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Theorem measbase 26776
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 5828 . 2  |-  ( M  e.  (measures `  S
)  ->  S  e.  dom measures )
2 vex 3081 . . . . 5  |-  s  e. 
_V
3 ovex 6228 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
4 mapex 7333 . . . . 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 3535 . . . 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 26775 . . 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 5651 . 2  |-  dom measures  =  U. ran sigAlgebra
111, 10syl6eleq 2552 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 1370    e. wcel 1758   {cab 2439   A.wral 2799   _Vcvv 3078   (/)c0 3748   ~Pcpw 3971   U.cuni 4202  Disj wdisj 4373   class class class wbr 4403   dom cdm 4951   ran crn 4952   -->wf 5525   ` cfv 5529  (class class class)co 6203   omcom 6589    ~<_ cdom 7421   0cc0 9396   +oocpnf 9529   [,]cicc 11417  Σ*cesum 26648  sigAlgebracsiga 26715  measurescmeas 26774
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-meas 26775
This theorem is referenced by:  measfrge0  26782  measvnul  26785  measvun  26788  measxun2  26789  measun  26790  measvuni  26793  measssd  26794  measunl  26795  measiuns  26796  measiun  26797  meascnbl  26798  measinblem  26799  measinb  26800  measinb2  26802  measdivcstOLD  26803  measdivcst  26804  aean  26824  mbfmbfm  26837  domprobsiga  26958  prob01  26960  probfinmeasbOLD  26975  probfinmeasb  26976  probmeasb  26977
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