Theorem unveldomd 34380

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 34376	. 2 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
3		sgon 34088	. 2 ⊢ (dom 𝑃 ∈ ∪ ran sigAlgebra → dom 𝑃 ∈ (sigAlgebra‘∪ dom 𝑃))
4		baselsiga 34079	. 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 4931  dom cdm 5700  ran crn 5701  ‘cfv 6573  sigAlgebracsiga 34072  Probcprb 34372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-esum 33992  df-siga 34073  df-meas 34160  df-prob 34373

This theorem is referenced by:  unveldom  34381  probdsb  34387  probtotrnd  34390  cndprobtot  34401  0rrv  34416  rrvadd  34417  dstfrvclim1  34442