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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 30802 | . 2 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴)) | |
3 | elrnsiga 30787 | . 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 2107 ∪ cuni 4671 ran crn 5356 ‘cfv 6135 sigAlgebracsiga 30768 sigaGencsigagen 30799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-int 4711 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-iota 6099 df-fun 6137 df-fv 6143 df-siga 30769 df-sigagen 30800 |
This theorem is referenced by: elsigagen2 30809 cldssbrsiga 30848 mbfmbfm 30918 imambfm 30922 sxbrsigalem2 30946 sxbrsiga 30950 sibf0 30994 sibff 30996 sibfinima 30999 sibfof 31000 sitgclg 31002 orvcval4 31121 orvcoel 31122 orvccel 31123 |
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