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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrnsiga | Structured version Visualization version GIF version |
Description: Dropping the base information off a sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.) |
Ref | Expression |
---|---|
elrnsiga | ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ ∪ ran sigAlgebra) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvssunirn 6940 | . 2 ⊢ (sigAlgebra‘𝑂) ⊆ ∪ ran sigAlgebra | |
2 | 1 | sseli 3991 | 1 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ ∪ ran sigAlgebra) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∪ cuni 4912 ran crn 5690 ‘cfv 6563 sigAlgebracsiga 34089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-cnv 5697 df-dm 5699 df-rn 5700 df-iota 6516 df-fv 6571 |
This theorem is referenced by: sgsiga 34123 sigapisys 34136 sigaldsys 34140 brsiga 34164 sxsiga 34172 measinb2 34204 pwcntmeas 34208 ddemeas 34217 cnmbfm 34245 elmbfmvol2 34249 mbfmcnt 34250 br2base 34251 dya2iocbrsiga 34257 dya2icobrsiga 34258 sxbrsiga 34272 omsmeas 34305 isrrvv 34425 rrvadd 34434 rrvmulc 34435 dstrvprob 34453 |
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