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| Description: The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.) | 
| Ref | Expression | 
|---|---|
| measbase | ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elfvdm 6943 | . 2 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ dom measures) | |
| 2 | vex 3484 | . . . . 5 ⊢ 𝑠 ∈ V | |
| 3 | ovex 7464 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
| 4 | mapex 7963 | . . . . 5 ⊢ ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V) | |
| 5 | 2, 3, 4 | mp2an 692 | . . . 4 ⊢ {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V | 
| 6 | simp1 1137 | . . . . 5 ⊢ ((𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦))) → 𝑚:𝑠⟶(0[,]+∞)) | |
| 7 | 6 | ss2abi 4067 | . . . 4 ⊢ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))} ⊆ {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} | 
| 8 | 5, 7 | ssexi 5322 | . . 3 ⊢ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))} ∈ V | 
| 9 | df-meas 34197 | . . 3 ⊢ 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)))}) | |
| 10 | 8, 9 | dmmpti 6712 | . 2 ⊢ dom measures = ∪ ran sigAlgebra | 
| 11 | 1, 10 | eleqtrdi 2851 | 1 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 Vcvv 3480 ∅c0 4333 𝒫 cpw 4600 ∪ cuni 4907 Disj wdisj 5110 class class class wbr 5143 dom cdm 5685 ran crn 5686 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ωcom 7887 ≼ cdom 8983 0cc0 11155 +∞cpnf 11292 [,]cicc 13390 Σ*cesum 34028 sigAlgebracsiga 34109 measurescmeas 34196 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-meas 34197 | 
| This theorem is referenced by: measfrge0 34204 measvnul 34207 measvun 34210 measxun2 34211 measun 34212 measvuni 34215 measssd 34216 measunl 34217 measiuns 34218 measiun 34219 meascnbl 34220 measinblem 34221 measinb 34222 measinb2 34224 measdivcst 34225 measdivcstALTV 34226 aean 34245 domprobsiga 34413 prob01 34415 probfinmeasb 34430 probfinmeasbALTV 34431 probmeasb 34432 | 
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