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Theorem elprob 33938
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 5896 . . . . 5 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
32unieqd 4915 . . . 4 (𝑝 = 𝑃 dom 𝑝 = dom 𝑃)
41, 3fveq12d 6891 . . 3 (𝑝 = 𝑃 → (𝑝 dom 𝑝) = (𝑃 dom 𝑃))
54eqeq1d 2728 . 2 (𝑝 = 𝑃 → ((𝑝 dom 𝑝) = 1 ↔ (𝑃 dom 𝑃) = 1))
6 df-prob 33937 . 2 Prob = {𝑝 ran measures ∣ (𝑝 dom 𝑝) = 1}
75, 6elrab2 3681 1 (𝑃 ∈ Prob ↔ (𝑃 ran measures ∧ (𝑃 dom 𝑃) = 1))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wcel 2098   cuni 4902  dom cdm 5669  ran crn 5670  cfv 6536  1c1 11110  measurescmeas 33723  Probcprb 33936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-dm 5679  df-iota 6488  df-fv 6544  df-prob 33937
This theorem is referenced by:  domprobmeas  33939  probtot  33941  probfinmeasb  33957  probfinmeasbALTV  33958  probmeasb  33959  dstrvprob  34000
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