Theorem elprob 34928

Description: The property of being a probability measure. (Contributed by Thierry Arnoux, 8-Dec-2016.)

Assertion

Ref	Expression
elprob	⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1))

Proof of Theorem elprob

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. . . 4 ⊢ (𝑝 = 𝑃 → 𝑝 = 𝑃)
2		dmeq 5891	. . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
3	2	unieqd 4883	. . . 4 ⊢ (𝑝 = 𝑃 → ∪ dom 𝑝 = ∪ dom 𝑃)
4	1, 3	fveq12d 6889	. . 3 ⊢ (𝑝 = 𝑃 → (𝑝‘∪ dom 𝑝) = (𝑃‘∪ dom 𝑃))
5	4	eqeq1d 2764	. 2 ⊢ (𝑝 = 𝑃 → ((𝑝‘∪ dom 𝑝) = 1 ↔ (𝑃‘∪ dom 𝑃) = 1))
6		df-prob 34927	. 2 ⊢ Prob = {𝑝 ∈ ∪ ran measures ∣ (𝑝‘∪ dom 𝑝) = 1}
7	5, 6	elrab2 3652	1 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∪ cuni 4870  dom cdm 5659  ran crn 5660  ‘cfv 6537  1c1 11129  measurescmeas 34714  Probcprb 34926

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-dm 5669  df-iota 6493  df-fv 6545  df-prob 34927

This theorem is used by:  domprobmeas  34929  probtot  34931  probfinmeasb  34947  probfinmeasbALTV  34948  probmeasb  34949  dstrvprob  34991