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Theorem domprobsiga 31743
Description: The domain of a probability measure is a sigma-algebra. (Contributed by Thierry Arnoux, 23-Dec-2016.)
Assertion
Ref Expression
domprobsiga (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)

Proof of Theorem domprobsiga
StepHypRef Expression
1 domprobmeas 31742 . 2 (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
2 measbase 31530 . 2 (𝑃 ∈ (measures‘dom 𝑃) → dom 𝑃 ran sigAlgebra)
31, 2syl 17 1 (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114   cuni 4813  dom cdm 5532  ran crn 5533  cfv 6334  sigAlgebracsiga 31441  measurescmeas 31528  Probcprb 31739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143  df-esum 31361  df-meas 31529  df-prob 31740
This theorem is referenced by:  unveldomd  31747  nuleldmp  31749  probdif  31752  totprobd  31758  cndprobin  31766  cndprob01  31767  isrrvv  31775  0rrv  31783  rrvadd  31784  rrvmulc  31785  orrvcval4  31796  orrvcoel  31797  orrvccel  31798
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