![]() |
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 | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
caragensspw.x | ⊢ 𝑋 = ∪ dom 𝑂 |
caragensspw.s | ⊢ 𝑆 = (CaraGen‘𝑂) |
Ref | Expression |
---|---|
caragensspw | ⊢ (𝜑 → 𝑆 ⊆ 𝒫 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragensspw.o | . . . 4 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
2 | caragensspw.s | . . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂) | |
3 | 2 | caragenss 41505 | . . . 4 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂) |
5 | pwuni 4696 | . . . 4 ⊢ dom 𝑂 ⊆ 𝒫 ∪ dom 𝑂 | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → dom 𝑂 ⊆ 𝒫 ∪ dom 𝑂) |
7 | 4, 6 | sstrd 3837 | . 2 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 ∪ dom 𝑂) |
8 | caragensspw.x | . . . . 5 ⊢ 𝑋 = ∪ dom 𝑂 | |
9 | 8 | pweqi 4382 | . . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂 |
10 | 9 | eqcomi 2834 | . . 3 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋 |
11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋) |
12 | 7, 11 | sseqtrd 3866 | 1 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 𝒫 cpw 4378 ∪ cuni 4658 dom cdm 5342 ‘cfv 6123 OutMeascome 41490 CaraGenccaragen 41492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-fv 6131 df-ov 6908 df-ome 41491 df-caragen 41493 |
This theorem is referenced by: caratheodorylem2 41528 |
Copyright terms: Public domain | W3C validator |