Theorem probmeasd 34607

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 34594	. . 3 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑃 ∈ (measures‘dom 𝑃))
4		measbasedom 34386	. 2 ⊢ (𝑃 ∈ ∪ ran measures ↔ 𝑃 ∈ (measures‘dom 𝑃))
5	3, 4	sylibr 235	1 ⊢ (𝜑 → 𝑃 ∈ ∪ ran measures)

Syntax hints:   → wi 4   ∈ wcel 2119  ∪ cuni 4838  dom cdm 5618  ran crn 5619  ‘cfv 6485  measurescmeas 34379  Probcprb 34591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-ov 7359  df-esum 34212  df-meas 34380  df-prob 34592

This theorem is referenced by: (None)