Theorem elprob 29798

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 5246	. . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
3	2	unieqd 4382	. . . 4 ⊢ (𝑝 = 𝑃 → ∪ dom 𝑝 = ∪ dom 𝑃)
4	1, 3	fveq12d 6109	. . 3 ⊢ (𝑝 = 𝑃 → (𝑝‘∪ dom 𝑝) = (𝑃‘∪ dom 𝑃))
5	4	eqeq1d 2612	. 2 ⊢ (𝑝 = 𝑃 → ((𝑝‘∪ dom 𝑝) = 1 ↔ (𝑃‘∪ dom 𝑃) = 1))
6		df-prob 29797	. 2 ⊢ Prob = {𝑝 ∈ ∪ ran measures ∣ (𝑝‘∪ dom 𝑝) = 1}
7	5, 6	elrab2 3333	1 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∪ cuni 4372  dom cdm 5038  ran crn 5039  ‘cfv 5804  1c1 9816  measurescmeas 29585  Probcprb 29796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812  df-prob 29797

This theorem is referenced by:  domprobmeas  29799  probtot  29801  probfinmeasbOLD  29817  probfinmeasb  29818  probmeasb  29819  dstrvprob  29860