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Theorem elprob 31777
 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 5736 . . . . 5 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
32unieqd 4814 . . . 4 (𝑝 = 𝑃 dom 𝑝 = dom 𝑃)
41, 3fveq12d 6652 . . 3 (𝑝 = 𝑃 → (𝑝 dom 𝑝) = (𝑃 dom 𝑃))
54eqeq1d 2800 . 2 (𝑝 = 𝑃 → ((𝑝 dom 𝑝) = 1 ↔ (𝑃 dom 𝑃) = 1))
6 df-prob 31776 . 2 Prob = {𝑝 ran measures ∣ (𝑝 dom 𝑝) = 1}
75, 6elrab2 3631 1 (𝑃 ∈ Prob ↔ (𝑃 ran measures ∧ (𝑃 dom 𝑃) = 1))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∪ cuni 4800  dom cdm 5519  ran crn 5520  ‘cfv 6324  1c1 10527  measurescmeas 31564  Probcprb 31775 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-dm 5529  df-iota 6283  df-fv 6332  df-prob 31776 This theorem is referenced by:  domprobmeas  31778  probtot  31780  probfinmeasb  31796  probfinmeasbALTV  31797  probmeasb  31798  dstrvprob  31839
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