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Mirrors > Home > MPE Home > Th. List > Mathboxes > sigagenid | Structured version Visualization version GIF version |
Description: The sigma-algebra generated by a sigma-algebra is itself. (Contributed by Thierry Arnoux, 4-Jun-2017.) |
Ref | Expression |
---|---|
sigagenid | ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (sigaGen‘𝑆) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sgon 33874 | . . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆)) | |
2 | ssid 3999 | . . 3 ⊢ 𝑆 ⊆ 𝑆 | |
3 | sigagenss 33899 | . . 3 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝑆) ∧ 𝑆 ⊆ 𝑆) → (sigaGen‘𝑆) ⊆ 𝑆) | |
4 | 1, 2, 3 | sylancl 584 | . 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (sigaGen‘𝑆) ⊆ 𝑆) |
5 | sssigagen 33895 | . 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ⊆ (sigaGen‘𝑆)) | |
6 | 4, 5 | eqssd 3994 | 1 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (sigaGen‘𝑆) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 ∪ cuni 4909 ran crn 5679 ‘cfv 6549 sigAlgebracsiga 33858 sigaGencsigagen 33888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-fv 6557 df-siga 33859 df-sigagen 33889 |
This theorem is referenced by: (None) |
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