Theorem probmeasd 34072

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 34059	. . 3 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑃 ∈ (measures‘dom 𝑃))
4		measbasedom 33850	. 2 ⊢ (𝑃 ∈ ∪ ran measures ↔ 𝑃 ∈ (measures‘dom 𝑃))
5	3, 4	sylibr 233	1 ⊢ (𝜑 → 𝑃 ∈ ∪ ran measures)

Syntax hints:   → wi 4   ∈ wcel 2098  ∪ cuni 4901  dom cdm 5670  ran crn 5671  ‘cfv 6541  measurescmeas 33843  Probcprb 34056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7417  df-esum 33676  df-meas 33844  df-prob 34057

This theorem is referenced by: (None)