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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 31775 | . 2 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴)) | |
3 | elrnsiga 31760 | . 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 2112 ∪ cuni 4805 ran crn 5537 ‘cfv 6358 sigAlgebracsiga 31742 sigaGencsigagen 31772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-int 4846 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fv 6366 df-siga 31743 df-sigagen 31773 |
This theorem is referenced by: elsigagen2 31782 cldssbrsiga 31821 mbfmbfm 31891 imambfm 31895 sxbrsigalem2 31919 sxbrsiga 31923 sibf0 31967 sibff 31969 sibfinima 31972 sibfof 31973 sitgclg 31975 orvcval4 32093 orvcoel 32094 orvccel 32095 |
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