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Theorem measbasedom 30495
 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 30493 . . . 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 482 . . 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 30494 . . . 4 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
4 ismeas 30492 . . . 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 247 . 2 (𝑀 ran measures → 𝑀 ∈ (measures‘dom 𝑀))
7 df-meas 30489 . . . 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 6040 . . 3 Fun measures
9 elunirn2 29681 . . 3 ((Fun measures ∧ 𝑀 ∈ (measures‘dom 𝑀)) → 𝑀 ran measures)
108, 9mpan 708 . 2 (𝑀 ∈ (measures‘dom 𝑀) → 𝑀 ran measures)
116, 10impbii 199 1 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1596   ∈ wcel 2103  {cab 2710  ∀wral 3014  ∅c0 4023  𝒫 cpw 4266  ∪ cuni 4544  Disj wdisj 4728   class class class wbr 4760  dom cdm 5218  ran crn 5219  Fun wfun 5995  ⟶wf 5997  ‘cfv 6001  (class class class)co 6765  ωcom 7182   ≼ cdom 8070  0cc0 10049  +∞cpnf 10184  [,]cicc 12292  Σ*cesum 30319  sigAlgebracsiga 30400  measurescmeas 30488 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-fv 6009  df-ov 6768  df-esum 30320  df-meas 30489 This theorem is referenced by:  truae  30536  aean  30537  mbfmbfm  30550  sibfinima  30631  sibfof  30632  domprobmeas  30702  probmeasd  30715  probfinmeasbOLD  30720  probfinmeasb  30721  probmeasb  30722  dstrvprob  30763
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