Theorem domprobmeas 31668

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 31667	. . 3 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1))
2	1	simplbi 500	. 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ ∪ ran measures)
3		measbasedom 31461	. 2 ⊢ (𝑃 ∈ ∪ ran measures ↔ 𝑃 ∈ (measures‘dom 𝑃))
4	2, 3	sylib 220	1 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ∪ cuni 4838  dom cdm 5555  ran crn 5556  ‘cfv 6355  1c1 10538  measurescmeas 31454  Probcprb 31665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-ov 7159  df-esum 31287  df-meas 31455  df-prob 31666

This theorem is referenced by:  domprobsiga  31669  prob01  31671  probnul  31672  probcun  31676  probinc  31679  probmeasd  31681  totprobd  31684  cndprob01  31693  cndprobprob  31696  dstrvprob  31729  dstfrvinc  31734  dstfrvclim1  31735