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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 29514 | . . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆)) | |
2 | ssid 3587 | . . 3 ⊢ 𝑆 ⊆ 𝑆 | |
3 | sigagenss 29539 | . . 3 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝑆) ∧ 𝑆 ⊆ 𝑆) → (sigaGen‘𝑆) ⊆ 𝑆) | |
4 | 1, 2, 3 | sylancl 693 | . 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (sigaGen‘𝑆) ⊆ 𝑆) |
5 | sssigagen 29535 | . 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ⊆ (sigaGen‘𝑆)) | |
6 | 4, 5 | eqssd 3585 | 1 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (sigaGen‘𝑆) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 ran crn 5039 ‘cfv 5804 sigAlgebracsiga 29497 sigaGencsigagen 29528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-siga 29498 df-sigagen 29529 |
This theorem is referenced by: (None) |
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