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Theorem elprob 28545
Description: The property of being a probability measure (Contributed by Thierry Arnoux, 8-Dec-2016.)
Assertion
Ref Expression
elprob  |-  ( P  e. Prob 
<->  ( P  e.  U. ran measures 
/\  ( P `  U. dom  P )  =  1 ) )

Proof of Theorem elprob
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4  |-  ( p  =  P  ->  p  =  P )
2 dmeq 5213 . . . . 5  |-  ( p  =  P  ->  dom  p  =  dom  P )
32unieqd 4261 . . . 4  |-  ( p  =  P  ->  U. dom  p  =  U. dom  P
)
41, 3fveq12d 5878 . . 3  |-  ( p  =  P  ->  (
p `  U. dom  p
)  =  ( P `
 U. dom  P
) )
54eqeq1d 2459 . 2  |-  ( p  =  P  ->  (
( p `  U. dom  p )  =  1  <-> 
( P `  U. dom  P )  =  1 ) )
6 df-prob 28544 . 2  |- Prob  =  {
p  e.  U. ran measures  |  ( p `  U. dom  p )  =  1 }
75, 6elrab2 3259 1  |-  ( P  e. Prob 
<->  ( P  e.  U. ran measures 
/\  ( P `  U. dom  P )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   U.cuni 4251   dom cdm 5008   ran crn 5009   ` cfv 5594   1c1 9510  measurescmeas 28339  Probcprb 28543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-dm 5018  df-iota 5557  df-fv 5602  df-prob 28544
This theorem is referenced by:  domprobmeas  28546  probtot  28548  probfinmeasbOLD  28564  probfinmeasb  28565  probmeasb  28566  dstrvprob  28607
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