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Theorem elprob 26722
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 5036 . . . . 5  |-  ( p  =  P  ->  dom  p  =  dom  P )
32unieqd 4098 . . . 4  |-  ( p  =  P  ->  U. dom  p  =  U. dom  P
)
41, 3fveq12d 5694 . . 3  |-  ( p  =  P  ->  (
p `  U. dom  p
)  =  ( P `
 U. dom  P
) )
54eqeq1d 2449 . 2  |-  ( p  =  P  ->  (
( p `  U. dom  p )  =  1  <-> 
( P `  U. dom  P )  =  1 ) )
6 df-prob 26721 . 2  |- Prob  =  {
p  e.  U. ran measures  |  ( p `  U. dom  p )  =  1 }
75, 6elrab2 3116 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 1364    e. wcel 1761   U.cuni 4088   dom cdm 4836   ran crn 4837   ` cfv 5415   1c1 9279  measurescmeas 26545  Probcprb 26720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-dm 4846  df-iota 5378  df-fv 5423  df-prob 26721
This theorem is referenced by:  domprobmeas  26723  probtot  26725  probfinmeasbOLD  26741  probfinmeasb  26742  probmeasb  26743  dstrvprob  26784
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