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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragenss | 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 domain of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragenss.1 | ⊢ 𝑆 = (CaraGen‘𝑂) |
Ref | Expression |
---|---|
caragenss | ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 4053 | . . 3 ⊢ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ⊆ 𝒫 ∪ dom 𝑂 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑂 ∈ OutMeas → {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ⊆ 𝒫 ∪ dom 𝑂) |
3 | caragenss.1 | . . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂) | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝑂 ∈ OutMeas → 𝑆 = (CaraGen‘𝑂)) |
5 | caragenval 42652 | . . . 4 ⊢ (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}) | |
6 | 4, 5 | eqtrd 2853 | . . 3 ⊢ (𝑂 ∈ OutMeas → 𝑆 = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}) |
7 | omedm 42658 | . . 3 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂) | |
8 | 6, 7 | sseq12d 3997 | . 2 ⊢ (𝑂 ∈ OutMeas → (𝑆 ⊆ dom 𝑂 ↔ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ⊆ 𝒫 ∪ dom 𝑂)) |
9 | 2, 8 | mpbird 258 | 1 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 ∖ cdif 3930 ∩ cin 3932 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 +𝑒 cxad 12493 OutMeascome 42648 CaraGenccaragen 42650 |
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 ax-un 7450 |
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-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-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-ome 42649 df-caragen 42651 |
This theorem is referenced by: caragensspw 42668 caragenuni 42670 caragendifcl 42673 caratheodorylem1 42685 caratheodorylem2 42686 dmvon 42765 voncmpl 42780 vonmblss 42799 |
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