Theorem domprobmeas 34661

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 34660	. . 3 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1))
2	1	simplbi 499	. 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ ∪ ran measures)
3		measbasedom 34453	. 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 1554   ∈ wcel 2136  ∪ cuni 4859  dom cdm 5640  ran crn 5641  ‘cfv 6510  1c1 11064  measurescmeas 34446  Probcprb 34658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-fv 6518  df-ov 7388  df-esum 34279  df-meas 34447  df-prob 34659

This theorem is referenced by:  domprobsiga  34662  prob01  34664  probnul  34665  probcun  34669  probinc  34672  probmeasd  34674  totprobd  34677  cndprob01  34686  cndprobprob  34689  boolesineq  34706  dstrvprob  34723  dstfrvinc  34728  dstfrvclim1  34729