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| Mirrors > Home > MPE Home > Th. List > Mathboxes > unisg | Structured version Visualization version GIF version | ||
| Description: The sigma-algebra generated by a collection 𝐴 is a sigma-algebra on ∪ 𝐴. (Contributed by Thierry Arnoux, 27-Dec-2016.) | 
| Ref | Expression | 
|---|---|
| unisg | ⊢ (𝐴 ∈ 𝑉 → ∪ (sigaGen‘𝐴) = ∪ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sigagensiga 34142 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴)) | |
| 2 | issgon 34124 | . . . 4 ⊢ ((sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴) ↔ ((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐴 = ∪ (sigaGen‘𝐴))) | |
| 3 | 1, 2 | sylib 218 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐴 = ∪ (sigaGen‘𝐴))) | 
| 4 | 3 | simprd 495 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (sigaGen‘𝐴)) | 
| 5 | 4 | eqcomd 2743 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ (sigaGen‘𝐴) = ∪ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cuni 4907 ran crn 5686 ‘cfv 6561 sigAlgebracsiga 34109 sigaGencsigagen 34139 | 
| 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-sbc 3789 df-csb 3900 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-int 4947 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-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-siga 34110 df-sigagen 34140 | 
| This theorem is referenced by: unibrsiga 34187 sxsigon 34193 imambfm 34264 cnmbfm 34265 sibf0 34336 sibff 34338 sibfof 34342 sitgclg 34344 orvcval4 34463 | 
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