Theorem measbasedom 33188

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isrnmeas 33186	. . . 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 33187	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
4		ismeas 33185	. . . 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		elfvunirn 6920	. 2 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → 𝑀 ∈ ∪ ran measures)
8	6, 7	impbii 208	1 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∅c0 4321  𝒫 cpw 4601  ∪ cuni 4907  Disj wdisj 5112   class class class wbr 5147  dom cdm 5675  ran crn 5676  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405  ωcom 7851   ≼ cdom 8933  0cc0 11106  +∞cpnf 11241  [,]cicc 13323  Σ^*cesum 33013  sigAlgebracsiga 33094  measurescmeas 33181

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-esum 33014  df-meas 33182

This theorem is referenced by:  truae  33229  aean  33230  sibfinima  33326  sibfof  33327  domprobmeas  33397  probmeasd  33410  probfinmeasb  33415  probfinmeasbALTV  33416  probmeasb  33417  dstrvprob  33458