Theorem dmmeas 33194

Description: The domain of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 19-Feb-2018.)

Assertion

Ref	Expression
dmmeas	⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)

Proof of Theorem dmmeas

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isrnmeas 33193	. 2 ⊢ (𝑀 ∈ ∪ ran measures → (dom 𝑀 ∈ ∪ ran sigAlgebra ∧ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)))))
2	1	simpld 495	1 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∅c0 4322  𝒫 cpw 4602  ∪ cuni 4908  Disj wdisj 5113   class class class wbr 5148  dom cdm 5676  ran crn 5677  ⟶wf 6539  ‘cfv 6543  (class class class)co 7408  ωcom 7854   ≼ cdom 8936  0cc0 11109  +∞cpnf 11244  [,]cicc 13326  Σ^*cesum 33020  sigAlgebracsiga 33101  measurescmeas 33188

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-esum 33021  df-meas 33189

This theorem is referenced by:  measbasedom  33195  aean  33237  sibf0  33328  sibff  33330  sibfinima  33333  sibfof  33334  sitgclg  33336