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 6692 | . 2 ⊢ (sigAlgebra‘𝑂) ⊆ ∪ ran sigAlgebra | |
2 | 1 | sseli 3960 | 1 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ ∪ ran sigAlgebra) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∪ cuni 4830 ran crn 5549 ‘cfv 6348 sigAlgebracsiga 31266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-cnv 5556 df-dm 5558 df-rn 5559 df-iota 6307 df-fv 6356 |
This theorem is referenced by: sgsiga 31300 sigapisys 31313 sigaldsys 31317 brsiga 31341 sxsiga 31349 measinb2 31381 pwcntmeas 31385 ddemeas 31394 cnmbfm 31420 elmbfmvol2 31424 mbfmcnt 31425 br2base 31426 dya2iocbrsiga 31432 dya2icobrsiga 31433 sxbrsiga 31447 omsmeas 31480 isrrvv 31600 rrvadd 31609 rrvmulc 31610 dstrvprob 31628 |
Copyright terms: Public domain | W3C validator |