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Theorem measbase 31444
Description: The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
measbase (𝑀 ∈ (measures‘𝑆) → 𝑆 ran sigAlgebra)

Proof of Theorem measbase
Dummy variables 𝑥 𝑚 𝑦 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6695 . 2 (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ dom measures)
2 vex 3496 . . . . 5 𝑠 ∈ V
3 ovex 7181 . . . . 5 (0[,]+∞) ∈ V
4 mapex 8404 . . . . 5 ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚𝑚:𝑠⟶(0[,]+∞)} ∈ V)
52, 3, 4mp2an 690 . . . 4 {𝑚𝑚:𝑠⟶(0[,]+∞)} ∈ V
6 simp1 1130 . . . . 5 ((𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦))) → 𝑚:𝑠⟶(0[,]+∞))
76ss2abi 4041 . . . 4 {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))} ⊆ {𝑚𝑚:𝑠⟶(0[,]+∞)}
85, 7ssexi 5217 . . 3 {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))} ∈ V
9 df-meas 31443 . . 3 measures = (𝑠 ran sigAlgebra ↦ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))})
108, 9dmmpti 6485 . 2 dom measures = ran sigAlgebra
111, 10eleqtrdi 2921 1 (𝑀 ∈ (measures‘𝑆) → 𝑆 ran sigAlgebra)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1081   = wceq 1530  wcel 2107  {cab 2797  wral 3136  Vcvv 3493  c0 4289  𝒫 cpw 4537   cuni 4830  Disj wdisj 5022   class class class wbr 5057  dom cdm 5548  ran crn 5549  wf 6344  cfv 6348  (class class class)co 7148  ωcom 7572  cdom 8499  0cc0 10529  +∞cpnf 10664  [,]cicc 12733  Σ*cesum 31274  sigAlgebracsiga 31355  measurescmeas 31442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151  df-meas 31443
This theorem is referenced by:  measfrge0  31450  measvnul  31453  measvun  31456  measxun2  31457  measun  31458  measvuni  31461  measssd  31462  measunl  31463  measiuns  31464  measiun  31465  meascnbl  31466  measinblem  31467  measinb  31468  measinb2  31470  measdivcst  31471  measdivcstALTV  31472  aean  31491  mbfmbfm  31504  domprobsiga  31657  prob01  31659  probfinmeasb  31674  probfinmeasbALTV  31675  probmeasb  31676
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