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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 31682 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴)) | |
2 | issgon 31664 | . . . 4 ⊢ ((sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴) ↔ ((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐴 = ∪ (sigaGen‘𝐴))) | |
3 | 1, 2 | sylib 221 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐴 = ∪ (sigaGen‘𝐴))) |
4 | 3 | simprd 499 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (sigaGen‘𝐴)) |
5 | 4 | eqcomd 2745 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ (sigaGen‘𝐴) = ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∪ cuni 4797 ran crn 5527 ‘cfv 6340 sigAlgebracsiga 31649 sigaGencsigagen 31679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-int 4838 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-fv 6348 df-siga 31650 df-sigagen 31680 |
This theorem is referenced by: unibrsiga 31727 sxsigon 31733 imambfm 31802 cnmbfm 31803 sibf0 31874 sibff 31876 sibfof 31880 sitgclg 31882 orvcval4 32000 |
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