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Theorem domprobsiga 32374
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 32373 . 2 (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
2 measbase 32161 . 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 2110   cuni 4845  dom cdm 5590  ran crn 5591  cfv 6432  sigAlgebracsiga 32072  measurescmeas 32159  Probcprb 32370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-fv 6440  df-ov 7274  df-esum 31992  df-meas 32160  df-prob 32371
This theorem is referenced by:  unveldomd  32378  nuleldmp  32380  probdif  32383  totprobd  32389  cndprobin  32397  cndprob01  32398  isrrvv  32406  0rrv  32414  rrvadd  32415  rrvmulc  32416  orrvcval4  32427  orrvcoel  32428  orrvccel  32429
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