|   | Mathbox for Thierry Arnoux | < Previous  
      Next > Nearby theorems | |
| 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 6939 | . 2 ⊢ (sigAlgebra‘𝑂) ⊆ ∪ ran sigAlgebra | |
| 2 | 1 | sseli 3979 | 1 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ ∪ ran sigAlgebra) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ∪ cuni 4907 ran crn 5686 ‘cfv 6561 sigAlgebracsiga 34109 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-iota 6514 df-fv 6569 | 
| This theorem is referenced by: sgsiga 34143 sigapisys 34156 sigaldsys 34160 brsiga 34184 sxsiga 34192 measinb2 34224 pwcntmeas 34228 ddemeas 34237 cnmbfm 34265 elmbfmvol2 34269 mbfmcnt 34270 br2base 34271 dya2iocbrsiga 34277 dya2icobrsiga 34278 sxbrsiga 34292 omsmeas 34325 isrrvv 34445 rrvadd 34454 rrvmulc 34455 dstrvprob 34474 | 
| Copyright terms: Public domain | W3C validator |