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Theorem elprob 34412
Description: The property of being a probability measure. (Contributed by Thierry Arnoux, 8-Dec-2016.)
Assertion
Ref Expression
elprob (𝑃 ∈ Prob ↔ (𝑃 ran measures ∧ (𝑃 dom 𝑃) = 1))

Proof of Theorem elprob
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 (𝑝 = 𝑃𝑝 = 𝑃)
2 dmeq 5913 . . . . 5 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
32unieqd 4919 . . . 4 (𝑝 = 𝑃 dom 𝑝 = dom 𝑃)
41, 3fveq12d 6912 . . 3 (𝑝 = 𝑃 → (𝑝 dom 𝑝) = (𝑃 dom 𝑃))
54eqeq1d 2738 . 2 (𝑝 = 𝑃 → ((𝑝 dom 𝑝) = 1 ↔ (𝑃 dom 𝑃) = 1))
6 df-prob 34411 . 2 Prob = {𝑝 ran measures ∣ (𝑝 dom 𝑝) = 1}
75, 6elrab2 3694 1 (𝑃 ∈ Prob ↔ (𝑃 ran measures ∧ (𝑃 dom 𝑃) = 1))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107   cuni 4906  dom cdm 5684  ran crn 5685  cfv 6560  1c1 11157  measurescmeas 34197  Probcprb 34410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-dm 5694  df-iota 6513  df-fv 6568  df-prob 34411
This theorem is referenced by:  domprobmeas  34413  probtot  34415  probfinmeasb  34431  probfinmeasbALTV  34432  probmeasb  34433  dstrvprob  34475
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