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Theorem domprobsiga 34393
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 34392 . 2 (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
2 measbase 34178 . 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 2106   cuni 4912  dom cdm 5689  ran crn 5690  cfv 6563  sigAlgebracsiga 34089  measurescmeas 34176  Probcprb 34389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-esum 34009  df-meas 34177  df-prob 34390
This theorem is referenced by:  unveldomd  34397  nuleldmp  34399  probdif  34402  totprobd  34408  cndprobin  34416  cndprob01  34417  isrrvv  34425  0rrv  34433  rrvadd  34434  rrvmulc  34435  orrvcval4  34446  orrvcoel  34447  orrvccel  34448
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