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Theorem unveldomd 30255
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 (𝜑𝑃 ∈ Prob)
Assertion
Ref Expression
unveldomd (𝜑 dom 𝑃 ∈ dom 𝑃)

Proof of Theorem unveldomd
StepHypRef Expression
1 unveldomd.1 . 2 (𝜑𝑃 ∈ Prob)
2 domprobsiga 30251 . 2 (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
3 sgon 29965 . 2 (dom 𝑃 ran sigAlgebra → dom 𝑃 ∈ (sigAlgebra‘ dom 𝑃))
4 baselsiga 29956 . 2 (dom 𝑃 ∈ (sigAlgebra‘ dom 𝑃) → dom 𝑃 ∈ dom 𝑃)
51, 2, 3, 44syl 19 1 (𝜑 dom 𝑃 ∈ dom 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987   cuni 4402  dom cdm 5074  ran crn 5075  cfv 5847  sigAlgebracsiga 29948  Probcprb 30247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-esum 29868  df-siga 29949  df-meas 30037  df-prob 30248
This theorem is referenced by:  unveldom  30256  probdsb  30262  probtotrnd  30265  cndprobtot  30276  0rrv  30291  rrvadd  30292  dstfrvclim1  30317
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