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Theorem elprob 34029
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 5906 . . . . 5 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
32unieqd 4921 . . . 4 (𝑝 = 𝑃 dom 𝑝 = dom 𝑃)
41, 3fveq12d 6904 . . 3 (𝑝 = 𝑃 → (𝑝 dom 𝑝) = (𝑃 dom 𝑃))
54eqeq1d 2730 . 2 (𝑝 = 𝑃 → ((𝑝 dom 𝑝) = 1 ↔ (𝑃 dom 𝑃) = 1))
6 df-prob 34028 . 2 Prob = {𝑝 ran measures ∣ (𝑝 dom 𝑝) = 1}
75, 6elrab2 3685 1 (𝑃 ∈ Prob ↔ (𝑃 ran measures ∧ (𝑃 dom 𝑃) = 1))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  wcel 2099   cuni 4908  dom cdm 5678  ran crn 5679  cfv 6548  1c1 11140  measurescmeas 33814  Probcprb 34027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-dm 5688  df-iota 6500  df-fv 6556  df-prob 34028
This theorem is referenced by:  domprobmeas  34030  probtot  34032  probfinmeasb  34048  probfinmeasbALTV  34049  probmeasb  34050  dstrvprob  34091
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