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Mathbox for Glauco Siliprandi |
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| 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 45797 | . . . 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 4942 | . . . 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 3987 | . 2 β’ (π β π β π« βͺ dom π) |
| 8 | caragensspw.x | . . . . 5 β’ π = βͺ dom π | |
| 9 | 8 | pweqi 4613 | . . . 4 β’ π« π = π« βͺ dom π |
| 10 | 9 | eqcomi 2735 | . . 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 4017 | 1 β’ (π β π β π« π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3943 π« cpw 4597 βͺ cuni 4902 dom cdm 5669 βcfv 6537 OutMeascome 45782 CaraGenccaragen 45784 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7408 df-ome 45783 df-caragen 45785 |
| This theorem is referenced by: caratheodorylem2 45820 |
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