Theorem probmeasd 34428

Description: A probability measure is a measure. (Contributed by Thierry Arnoux, 2-Feb-2017.)

Hypothesis

Ref	Expression
probmeasd.1	⊢ (𝜑 → 𝑃 ∈ Prob)

Assertion

Ref	Expression
probmeasd	⊢ (𝜑 → 𝑃 ∈ ∪ ran measures)

Proof of Theorem probmeasd

Step	Hyp	Ref	Expression
1		probmeasd.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ Prob)
2		domprobmeas 34415	. . 3 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑃 ∈ (measures‘dom 𝑃))
4		measbasedom 34207	. 2 ⊢ (𝑃 ∈ ∪ ran measures ↔ 𝑃 ∈ (measures‘dom 𝑃))
5	3, 4	sylibr 234	1 ⊢ (𝜑 → 𝑃 ∈ ∪ ran measures)

Syntax hints:   → wi 4   ∈ wcel 2111  ∪ cuni 4854  dom cdm 5611  ran crn 5612  ‘cfv 6476  measurescmeas 34200  Probcprb 34412

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-ov 7344  df-esum 34033  df-meas 34201  df-prob 34413

This theorem is referenced by: (None)