Nearby theorems
|
|
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > caragensspw | Structured version Visualization version GIF version | ||
| Description: The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the power set of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| Ref | Expression |
|---|---|
| caragensspw.o | β’ (π β π β OutMeas) |
| caragensspw.x | β’ π = βͺ dom π |
| caragensspw.s | β’ π = (CaraGenβπ) |
| Ref | Expression |
|---|---|
| caragensspw | β’ (π β π β π« π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | caragensspw.o | . . . 4 β’ (π β π β OutMeas) | |
| 2 | caragensspw.s | . . . . 5 β’ π = (CaraGenβπ) | |
| 3 | 2 | caragenss 45206 | . . . 4 β’ (π β OutMeas β π β dom π) |
| 4 | 1, 3 | syl 17 | . . 3 β’ (π β π β dom π) |
| 5 | pwuni 4948 | . . . 4 β’ dom π β π« βͺ dom π | |
| 6 | 5 | a1i 11 | . . 3 β’ (π β dom π β π« βͺ dom π) |
| 7 | 4, 6 | sstrd 3991 | . 2 β’ (π β π β π« βͺ dom π) |
| 8 | caragensspw.x | . . . . 5 β’ π = βͺ dom π | |
| 9 | 8 | pweqi 4617 | . . . 4 β’ π« π = π« βͺ dom π |
| 10 | 9 | eqcomi 2741 | . . 3 β’ π« βͺ dom π = π« π |
| 11 | 10 | a1i 11 | . 2 β’ (π β π« βͺ dom π = π« π) |
| 12 | 7, 11 | sseqtrd 4021 | 1 β’ (π β π β π« π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1541 β wcel 2106 β wss 3947 π« cpw 4601 βͺ cuni 4907 dom cdm 5675 βcfv 6540 OutMeascome 45191 CaraGenccaragen 45193 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7408 df-ome 45192 df-caragen 45194 |
| This theorem is referenced by: caratheodorylem2 45229 |
| Copyright terms: Public domain | W3C validator |