Theorem probmeasd 34380

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 34367	. . 3 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑃 ∈ (measures‘dom 𝑃))
4		measbasedom 34158	. 2 ⊢ (𝑃 ∈ ∪ ran measures ↔ 𝑃 ∈ (measures‘dom 𝑃))
5	3, 4	sylibr 234	1 ⊢ (𝜑 → 𝑃 ∈ ∪ ran measures)

Syntax hints:   → wi 4   ∈ wcel 2108  ∪ cuni 4931  dom cdm 5695  ran crn 5696  ‘cfv 6568  measurescmeas 34151  Probcprb 34364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7764

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-iota 6520  df-fun 6570  df-fn 6571  df-f 6572  df-fv 6576  df-ov 7446  df-esum 33984  df-meas 34152  df-prob 34365

This theorem is referenced by: (None)