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Theorem dmmeas 33130
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 33129 . 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 3062  c0 4320  𝒫 cpw 4598   cuni 4904  Disj wdisj 5109   class class class wbr 5144  dom cdm 5672  ran crn 5673  wf 6531  cfv 6535  (class class class)co 7396  ωcom 7842  cdom 8925  0cc0 11097  +∞cpnf 11232  [,]cicc 13314  Σ*cesum 32956  sigAlgebracsiga 33037  measurescmeas 33124
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 2704  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-fv 6543  df-ov 7399  df-esum 32957  df-meas 33125
This theorem is referenced by:  measbasedom  33131  aean  33173  sibf0  33264  sibff  33266  sibfinima  33269  sibfof  33270  sitgclg  33272
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