Theorem unveldomd 34379

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 34375	. 2 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
3		sgon 34087	. 2 ⊢ (dom 𝑃 ∈ ∪ ran sigAlgebra → dom 𝑃 ∈ (sigAlgebra‘∪ dom 𝑃))
4		baselsiga 34078	. 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 2109  ∪ cuni 4867  dom cdm 5631  ran crn 5632  ‘cfv 6499  sigAlgebracsiga 34071  Probcprb 34371

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7372  df-esum 33991  df-siga 34072  df-meas 34159  df-prob 34372

This theorem is referenced by:  unveldom  34380  probdsb  34386  probtotrnd  34389  cndprobtot  34400  0rrv  34415  rrvadd  34416  dstfrvclim1  34442