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Theorem elprob 33049
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 5864 . . . . 5 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
32unieqd 4884 . . . 4 (𝑝 = 𝑃 dom 𝑝 = dom 𝑃)
41, 3fveq12d 6854 . . 3 (𝑝 = 𝑃 → (𝑝 dom 𝑝) = (𝑃 dom 𝑃))
54eqeq1d 2739 . 2 (𝑝 = 𝑃 → ((𝑝 dom 𝑝) = 1 ↔ (𝑃 dom 𝑃) = 1))
6 df-prob 33048 . 2 Prob = {𝑝 ran measures ∣ (𝑝 dom 𝑝) = 1}
75, 6elrab2 3653 1 (𝑃 ∈ Prob ↔ (𝑃 ran measures ∧ (𝑃 dom 𝑃) = 1))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107   cuni 4870  dom cdm 5638  ran crn 5639  cfv 6501  1c1 11059  measurescmeas 32834  Probcprb 33047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-dm 5648  df-iota 6453  df-fv 6509  df-prob 33048
This theorem is referenced by:  domprobmeas  33050  probtot  33052  probfinmeasb  33068  probfinmeasbALTV  33069  probmeasb  33070  dstrvprob  33111
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