Theorem dmmeas 31570

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 31569	. 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 498	1 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∅c0 4243  𝒫 cpw 4497  ∪ cuni 4800  Disj wdisj 4995   class class class wbr 5030  dom cdm 5519  ran crn 5520  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  ωcom 7560   ≼ cdom 8490  0cc0 10526  +∞cpnf 10661  [,]cicc 12729  Σ^*cesum 31396  sigAlgebracsiga 31477  measurescmeas 31564

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-esum 31397  df-meas 31565

This theorem is referenced by:  measbasedom  31571  aean  31613  sibf0  31702  sibff  31704  sibfinima  31707  sibfof  31708  sitgclg  31710