Theorem dmmeas 30604

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 30603	. 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 482	1 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)

Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∅c0 4063  𝒫 cpw 4297  ∪ cuni 4574  Disj wdisj 4754   class class class wbr 4786  dom cdm 5249  ran crn 5250  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793  ωcom 7212   ≼ cdom 8107  0cc0 10138  +∞cpnf 10273  [,]cicc 12383  Σ^*cesum 30429  sigAlgebracsiga 30510  measurescmeas 30598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-esum 30430  df-meas 30599

This theorem is referenced by:  measbasedom  30605  aean  30647  sibf0  30736  sibff  30738  sibfinima  30741  sibfof  30742  sitgclg  30744