Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > sigagenss | Structured version Visualization version GIF version |
Description: The generated sigma-algebra is a subset of all sigma-algebras containing the generating set, i.e. the generated sigma-algebra is the smallest sigma-algebra containing the generating set, here 𝐴. (Contributed by Thierry Arnoux, 4-Jun-2017.) |
Ref | Expression |
---|---|
sigagenss | ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → (sigaGen‘𝐴) ⊆ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexg 5219 | . . . 4 ⊢ ((𝐴 ⊆ 𝑆 ∧ 𝑆 ∈ (sigAlgebra‘∪ 𝐴)) → 𝐴 ∈ V) | |
2 | 1 | ancoms 461 | . . 3 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → 𝐴 ∈ V) |
3 | sigagenval 31394 | . . 3 ⊢ (𝐴 ∈ V → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠}) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠}) |
5 | sseq2 3992 | . . 3 ⊢ (𝑠 = 𝑆 → (𝐴 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝑆)) | |
6 | 5 | intminss 4894 | . 2 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ 𝑆) |
7 | 4, 6 | eqsstrd 4004 | 1 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → (sigaGen‘𝐴) ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 ⊆ wss 3935 ∪ cuni 4831 ∩ cint 4868 ‘cfv 6349 sigAlgebracsiga 31362 sigaGencsigagen 31392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-siga 31363 df-sigagen 31393 |
This theorem is referenced by: sigagenss2 31404 sigagenid 31405 imambfm 31515 |
Copyright terms: Public domain | W3C validator |