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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 5227 | . . . 4 ⊢ ((𝐴 ⊆ 𝑆 ∧ 𝑆 ∈ (sigAlgebra‘∪ 𝐴)) → 𝐴 ∈ V) | |
2 | 1 | ancoms 461 | . . 3 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → 𝐴 ∈ V) |
3 | sigagenval 31399 | . . 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 3993 | . . 3 ⊢ (𝑠 = 𝑆 → (𝐴 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝑆)) | |
6 | 5 | intminss 4902 | . 2 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ 𝑆) |
7 | 4, 6 | eqsstrd 4005 | 1 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → (sigaGen‘𝐴) ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3142 Vcvv 3494 ⊆ wss 3936 ∪ cuni 4838 ∩ cint 4876 ‘cfv 6355 sigAlgebracsiga 31367 sigaGencsigagen 31397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-siga 31368 df-sigagen 31398 |
This theorem is referenced by: sigagenss2 31409 sigagenid 31410 imambfm 31520 |
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