Theorem domprobmeas 34434

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 34433	. . 3 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1))
2	1	simplbi 497	. 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ ∪ ran measures)
3		measbasedom 34226	. 2 ⊢ (𝑃 ∈ ∪ ran measures ↔ 𝑃 ∈ (measures‘dom 𝑃))
4	2, 3	sylib 218	1 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ∪ cuni 4860  dom cdm 5621  ran crn 5622  ‘cfv 6489  1c1 11017  measurescmeas 34219  Probcprb 34431

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-ov 7358  df-esum 34052  df-meas 34220  df-prob 34432

This theorem is referenced by:  domprobsiga  34435  prob01  34437  probnul  34438  probcun  34442  probinc  34445  probmeasd  34447  totprobd  34450  cndprob01  34459  cndprobprob  34462  boolesineq  34479  dstrvprob  34496  dstfrvinc  34501  dstfrvclim1  34502