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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sgsiga | Structured version Visualization version GIF version |
Description: A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
Ref | Expression |
---|---|
sgsiga.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
sgsiga | ⊢ (𝜑 → (sigaGen‘𝐴) ∈ ∪ ran sigAlgebra) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sgsiga.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | sigagensiga 33974 | . 2 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴)) | |
3 | elrnsiga 33959 | . 2 ⊢ ((sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴) → (sigaGen‘𝐴) ∈ ∪ ran sigAlgebra) | |
4 | 1, 2, 3 | 3syl 18 | 1 ⊢ (𝜑 → (sigaGen‘𝐴) ∈ ∪ ran sigAlgebra) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ∪ cuni 4913 ran crn 5683 ‘cfv 6554 sigAlgebracsiga 33941 sigaGencsigagen 33971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6506 df-fun 6556 df-fv 6562 df-siga 33942 df-sigagen 33972 |
This theorem is referenced by: elsigagen2 33981 cldssbrsiga 34020 imambfm 34096 sxbrsigalem2 34120 sxbrsiga 34124 sibf0 34168 sibff 34170 sibfinima 34173 sibfof 34174 sitgclg 34176 orvcval4 34294 orvcoel 34295 orvccel 34296 |
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