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Theorem dmmeas 32840
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.
StepHypRef Expression
1 isrnmeas 32839 . 2 (𝑀 ran measures → (dom 𝑀 ran sigAlgebra ∧ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
21simpld 496 1 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3065  c0 4287  𝒫 cpw 4565   cuni 4870  Disj wdisj 5075   class class class wbr 5110  dom cdm 5638  ran crn 5639  wf 6497  cfv 6501  (class class class)co 7362  ωcom 7807  cdom 8888  0cc0 11058  +∞cpnf 11193  [,]cicc 13274  Σ*cesum 32666  sigAlgebracsiga 32747  measurescmeas 32834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-esum 32667  df-meas 32835
This theorem is referenced by:  measbasedom  32841  aean  32883  sibf0  32974  sibff  32976  sibfinima  32979  sibfof  32980  sitgclg  32982
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