Theorem unveldomd 33352

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 33348	. 2 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
3		sgon 33060	. 2 ⊢ (dom 𝑃 ∈ ∪ ran sigAlgebra → dom 𝑃 ∈ (sigAlgebra‘∪ dom 𝑃))
4		baselsiga 33051	. 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 2107  ∪ cuni 4907  dom cdm 5675  ran crn 5676  ‘cfv 6540  sigAlgebracsiga 33044  Probcprb 33344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7407  df-esum 32964  df-siga 33045  df-meas 33132  df-prob 33345

This theorem is referenced by:  unveldom  33353  probdsb  33359  probtotrnd  33362  cndprobtot  33373  0rrv  33388  rrvadd  33389  dstfrvclim1  33414