Theorem domprobmeas 32377

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 32376	. . 3 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1))
2	1	simplbi 498	. 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ ∪ ran measures)
3		measbasedom 32170	. 2 ⊢ (𝑃 ∈ ∪ ran measures ↔ 𝑃 ∈ (measures‘dom 𝑃))
4	2, 3	sylib 217	1 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∪ cuni 4839  dom cdm 5589  ran crn 5590  ‘cfv 6433  1c1 10872  measurescmeas 32163  Probcprb 32374

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-esum 31996  df-meas 32164  df-prob 32375

This theorem is referenced by:  domprobsiga  32378  prob01  32380  probnul  32381  probcun  32385  probinc  32388  probmeasd  32390  totprobd  32393  cndprob01  32402  cndprobprob  32405  dstrvprob  32438  dstfrvinc  32443  dstfrvclim1  32444