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Theorem unveldomd 30817
 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 30813 . 2 (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
3 sgon 30527 . 2 (dom 𝑃 ran sigAlgebra → dom 𝑃 ∈ (sigAlgebra‘ dom 𝑃))
4 baselsiga 30518 . 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 2145  ∪ cuni 4574  dom cdm 5249  ran crn 5250  ‘cfv 6031  sigAlgebracsiga 30510  Probcprb 30809 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-esum 30430  df-siga 30511  df-meas 30599  df-prob 30810 This theorem is referenced by:  unveldom  30818  probdsb  30824  probtotrnd  30827  cndprobtot  30838  0rrv  30853  rrvadd  30854  dstfrvclim1  30879
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