Theorem unveldomd 34417

Description: The universe is an element of the domain of the probability, the universe (entire probability space) being ∪ dom 𝑃 in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)

Hypothesis

Ref	Expression
unveldomd.1	⊢ (𝜑 → 𝑃 ∈ Prob)

Assertion

Ref	Expression
unveldomd	⊢ (𝜑 → ∪ dom 𝑃 ∈ dom 𝑃)

Proof of Theorem unveldomd

Step	Hyp	Ref	Expression
1		unveldomd.1	. 2 ⊢ (𝜑 → 𝑃 ∈ Prob)
2		domprobsiga 34413	. 2 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
3		sgon 34125	. 2 ⊢ (dom 𝑃 ∈ ∪ ran sigAlgebra → dom 𝑃 ∈ (sigAlgebra‘∪ dom 𝑃))
4		baselsiga 34116	. 2 ⊢ (dom 𝑃 ∈ (sigAlgebra‘∪ dom 𝑃) → ∪ dom 𝑃 ∈ dom 𝑃)
5	1, 2, 3, 4	4syl 19	1 ⊢ (𝜑 → ∪ dom 𝑃 ∈ dom 𝑃)

Syntax hints:   → wi 4   ∈ wcel 2108  ∪ cuni 4907  dom cdm 5685  ran crn 5686  ‘cfv 6561  sigAlgebracsiga 34109  Probcprb 34409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-esum 34029  df-siga 34110  df-meas 34197  df-prob 34410

This theorem is referenced by:  unveldom  34418  probdsb  34424  probtotrnd  34427  cndprobtot  34438  0rrv  34453  rrvadd  34454  dstfrvclim1  34480