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Theorem probmeasd 30819
 Description: A probability measure is a measure. (Contributed by Thierry Arnoux, 2-Feb-2017.)
Hypothesis
Ref Expression
probmeasd.1 (𝜑𝑃 ∈ Prob)
Assertion
Ref Expression
probmeasd (𝜑𝑃 ran measures)

Proof of Theorem probmeasd
StepHypRef Expression
1 probmeasd.1 . . 3 (𝜑𝑃 ∈ Prob)
2 domprobmeas 30806 . . 3 (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
31, 2syl 17 . 2 (𝜑𝑃 ∈ (measures‘dom 𝑃))
4 measbasedom 30599 . 2 (𝑃 ran measures ↔ 𝑃 ∈ (measures‘dom 𝑃))
53, 4sylibr 224 1 (𝜑𝑃 ran measures)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2144  ∪ cuni 4572  dom cdm 5249  ran crn 5250  ‘cfv 6031  measurescmeas 30592  Probcprb 30803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-esum 30424  df-meas 30593  df-prob 30804 This theorem is referenced by: (None)
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