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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 31402 | . 2 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴)) | |
3 | elrnsiga 31387 | . 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 2114 ∪ cuni 4840 ran crn 5558 ‘cfv 6357 sigAlgebracsiga 31369 sigaGencsigagen 31399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fv 6365 df-siga 31370 df-sigagen 31400 |
This theorem is referenced by: elsigagen2 31409 cldssbrsiga 31448 mbfmbfm 31518 imambfm 31522 sxbrsigalem2 31546 sxbrsiga 31550 sibf0 31594 sibff 31596 sibfinima 31599 sibfof 31600 sitgclg 31602 orvcval4 31720 orvcoel 31721 orvccel 31722 |
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