Theorem domprobsiga 34496

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

Step	Hyp	Ref	Expression
1		domprobmeas 34495	. 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
2		measbase 34282	. 2 ⊢ (𝑃 ∈ (measures‘dom 𝑃) → dom 𝑃 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 17	1 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)

Syntax hints:   → wi 4   ∈ wcel 2113  ∪ cuni 4860  dom cdm 5621  ran crn 5622  ‘cfv 6489  sigAlgebracsiga 34193  measurescmeas 34280  Probcprb 34492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-ov 7358  df-esum 34113  df-meas 34281  df-prob 34493

This theorem is referenced by:  unveldomd  34500  nuleldmp  34502  probdif  34505  totprobd  34511  cndprobin  34519  cndprob01  34520  isrrvv  34528  0rrv  34536  rrvadd  34537  rrvmulc  34538  boolesineq  34540  orrvcval4  34550  orrvcoel  34551  orrvccel  34552