Theorem domprobsiga 34571

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 34570	. 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
2		measbase 34357	. 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 2114  ∪ cuni 4851  dom cdm 5624  ran crn 5625  ‘cfv 6492  sigAlgebracsiga 34268  measurescmeas 34355  Probcprb 34567

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7363  df-esum 34188  df-meas 34356  df-prob 34568

This theorem is referenced by:  unveldomd  34575  nuleldmp  34577  probdif  34580  totprobd  34586  cndprobin  34594  cndprob01  34595  isrrvv  34603  0rrv  34611  rrvadd  34612  rrvmulc  34613  boolesineq  34615  orrvcval4  34625  orrvcoel  34626  orrvccel  34627