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Theorem measbasedom 31575
Description: The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
measbasedom (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))

Proof of Theorem measbasedom
Dummy variables 𝑥 𝑦 𝑚 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isrnmeas 31573 . . . 4 (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)))))
21simprd 499 . . 3 (𝑀 ran measures → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))))
3 dmmeas 31574 . . . 4 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
4 ismeas 31572 . . . 4 (dom 𝑀 ran sigAlgebra → (𝑀 ∈ (measures‘dom 𝑀) ↔ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
53, 4syl 17 . . 3 (𝑀 ran measures → (𝑀 ∈ (measures‘dom 𝑀) ↔ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
62, 5mpbird 260 . 2 (𝑀 ran measures → 𝑀 ∈ (measures‘dom 𝑀))
7 df-meas 31569 . . . 4 measures = (𝑠 ran sigAlgebra ↦ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))})
87funmpt2 6367 . . 3 Fun measures
9 elunirn2 30418 . . 3 ((Fun measures ∧ 𝑀 ∈ (measures‘dom 𝑀)) → 𝑀 ran measures)
108, 9mpan 689 . 2 (𝑀 ∈ (measures‘dom 𝑀) → 𝑀 ran measures)
116, 10impbii 212 1 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  {cab 2779  wral 3109  c0 4246  𝒫 cpw 4500   cuni 4803  Disj wdisj 4998   class class class wbr 5033  dom cdm 5523  ran crn 5524  Fun wfun 6322  wf 6324  cfv 6328  (class class class)co 7139  ωcom 7564  cdom 8494  0cc0 10530  +∞cpnf 10665  [,]cicc 12733  Σ*cesum 31400  sigAlgebracsiga 31481  measurescmeas 31568
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-esum 31401  df-meas 31569
This theorem is referenced by:  truae  31616  aean  31617  mbfmbfm  31630  sibfinima  31711  sibfof  31712  domprobmeas  31782  probmeasd  31795  probfinmeasb  31800  probfinmeasbALTV  31801  probmeasb  31802  dstrvprob  31843
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