Theorem measbasedom 32170

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.

Step	Hyp	Ref	Expression
1		isrnmeas 32168	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → (dom 𝑀 ∈ ∪ ran sigAlgebra ∧ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)))))
2	1	simprd 496	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))))
3		dmmeas 32169	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
4		ismeas 32167	. . . 4 ⊢ (dom 𝑀 ∈ ∪ ran sigAlgebra → (𝑀 ∈ (measures‘dom 𝑀) ↔ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)))))
5	3, 4	syl 17	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → (𝑀 ∈ (measures‘dom 𝑀) ↔ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)))))
6	2, 5	mpbird 256	. 2 ⊢ (𝑀 ∈ ∪ ran measures → 𝑀 ∈ (measures‘dom 𝑀))
7		df-meas 32164	. . . 4 ⊢ measures = (𝑠 ∈ ∪ ran sigAlgebra ↦ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))})
8	7	funmpt2 6473	. . 3 ⊢ Fun measures
9		elunirn2 7126	. . 3 ⊢ ((Fun measures ∧ 𝑀 ∈ (measures‘dom 𝑀)) → 𝑀 ∈ ∪ ran measures)
10	8, 9	mpan 687	. 2 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → 𝑀 ∈ ∪ ran measures)
11	6, 10	impbii 208	1 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∅c0 4256  𝒫 cpw 4533  ∪ cuni 4839  Disj wdisj 5039   class class class wbr 5074  dom cdm 5589  ran crn 5590  Fun wfun 6427  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  ωcom 7712   ≼ cdom 8731  0cc0 10871  +∞cpnf 11006  [,]cicc 13082  Σ^*cesum 31995  sigAlgebracsiga 32076  measurescmeas 32163

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-esum 31996  df-meas 32164

This theorem is referenced by:  truae  32211  aean  32212  mbfmbfm  32225  sibfinima  32306  sibfof  32307  domprobmeas  32377  probmeasd  32390  probfinmeasb  32395  probfinmeasbALTV  32396  probmeasb  32397  dstrvprob  32438