Theorem elprob 34391

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 22	. . . 4 ⊢ (𝑝 = 𝑃 → 𝑝 = 𝑃)
2		dmeq 5917	. . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
3	2	unieqd 4925	. . . 4 ⊢ (𝑝 = 𝑃 → ∪ dom 𝑝 = ∪ dom 𝑃)
4	1, 3	fveq12d 6914	. . 3 ⊢ (𝑝 = 𝑃 → (𝑝‘∪ dom 𝑝) = (𝑃‘∪ dom 𝑃))
5	4	eqeq1d 2737	. 2 ⊢ (𝑝 = 𝑃 → ((𝑝‘∪ dom 𝑝) = 1 ↔ (𝑃‘∪ dom 𝑃) = 1))
6		df-prob 34390	. 2 ⊢ Prob = {𝑝 ∈ ∪ ran measures ∣ (𝑝‘∪ dom 𝑝) = 1}
7	5, 6	elrab2 3698	1 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∪ cuni 4912  dom cdm 5689  ran crn 5690  ‘cfv 6563  1c1 11154  measurescmeas 34176  Probcprb 34389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571  df-prob 34390

This theorem is referenced by:  domprobmeas  34392  probtot  34394  probfinmeasb  34410  probfinmeasbALTV  34411  probmeasb  34412  dstrvprob  34453