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Theorem probmeasd 34403
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 34390 . . 3 (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
31, 2syl 17 . 2 (𝜑𝑃 ∈ (measures‘dom 𝑃))
4 measbasedom 34181 . 2 (𝑃 ran measures ↔ 𝑃 ∈ (measures‘dom 𝑃))
53, 4sylibr 234 1 (𝜑𝑃 ran measures)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108   cuni 4905  dom cdm 5683  ran crn 5684  cfv 6559  measurescmeas 34174  Probcprb 34387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-fv 6567  df-ov 7432  df-esum 34007  df-meas 34175  df-prob 34388
This theorem is referenced by: (None)
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