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Theorem unveldomd 31076
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 31072 . 2 (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
3 sgon 30785 . 2 (dom 𝑃 ran sigAlgebra → dom 𝑃 ∈ (sigAlgebra‘ dom 𝑃))
4 baselsiga 30776 . 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 2107   cuni 4671  dom cdm 5355  ran crn 5356  cfv 6135  sigAlgebracsiga 30768  Probcprb 31068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-esum 30688  df-siga 30769  df-meas 30857  df-prob 31069
This theorem is referenced by:  unveldom  31077  probdsb  31083  probtotrnd  31086  cndprobtot  31097  0rrv  31112  rrvadd  31113  dstfrvclim1  31138
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