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Theorem elprob 26792
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 5040 . . . . 5  |-  ( p  =  P  ->  dom  p  =  dom  P )
32unieqd 4101 . . . 4  |-  ( p  =  P  ->  U. dom  p  =  U. dom  P
)
41, 3fveq12d 5697 . . 3  |-  ( p  =  P  ->  (
p `  U. dom  p
)  =  ( P `
 U. dom  P
) )
54eqeq1d 2451 . 2  |-  ( p  =  P  ->  (
( p `  U. dom  p )  =  1  <-> 
( P `  U. dom  P )  =  1 ) )
6 df-prob 26791 . 2  |- Prob  =  {
p  e.  U. ran measures  |  ( p `  U. dom  p )  =  1 }
75, 6elrab2 3119 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 1369    e. wcel 1756   U.cuni 4091   dom cdm 4840   ran crn 4841   ` cfv 5418   1c1 9283  measurescmeas 26609  Probcprb 26790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-dm 4850  df-iota 5381  df-fv 5426  df-prob 26791
This theorem is referenced by:  domprobmeas  26793  probtot  26795  probfinmeasbOLD  26811  probfinmeasb  26812  probmeasb  26813  dstrvprob  26854
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