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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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