Theorem domprobmeas 34923

Description: A probability measure is a measure on its domain. (Contributed by Thierry Arnoux, 23-Dec-2016.)

Assertion

Ref	Expression
domprobmeas	⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))

Proof of Theorem domprobmeas

Step	Hyp	Ref	Expression
1		elprob 34922	. . 3 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1))
2	1	simplbi 502	. 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ ∪ ran measures)
3		measbasedom 34715	. 2 ⊢ (𝑃 ∈ ∪ ran measures ↔ 𝑃 ∈ (measures‘dom 𝑃))
4	2, 3	sylib 221	1 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∪ cuni 4870  dom cdm 5659  ran crn 5660  ‘cfv 6537  1c1 11128  measurescmeas 34708  Probcprb 34920

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7419  df-esum 34540  df-meas 34709  df-prob 34921

This theorem is used by:  domprobsiga  34924  prob01  34926  probnul  34927  probcun  34931  probinc  34934  probmeasd  34936  totprobd  34939  cndprob01  34948  cndprobprob  34951  boolesineq  34968  dstrvprob  34985  dstfrvinc  34990  dstfrvclim1  34991