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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 44819 | . . . 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 4911 | . . . 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 3959 | . 2 β’ (π β π β π« βͺ dom π) |
8 | caragensspw.x | . . . . 5 β’ π = βͺ dom π | |
9 | 8 | pweqi 4581 | . . . 4 β’ π« π = π« βͺ dom π |
10 | 9 | eqcomi 2746 | . . 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 3989 | 1 β’ (π β π β π« π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3915 π« cpw 4565 βͺ cuni 4870 dom cdm 5638 βcfv 6501 OutMeascome 44804 CaraGenccaragen 44806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-ov 7365 df-ome 44805 df-caragen 44807 |
This theorem is referenced by: caratheodorylem2 44842 |
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