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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragenelss | Structured version Visualization version GIF version |
Description: An element of the Caratheodory's construction is a subset of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragenelss.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
caragenelss.s | ⊢ 𝑆 = (CaraGen‘𝑂) |
caragenelss.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
caragenelss.x | ⊢ 𝑋 = ∪ dom 𝑂 |
Ref | Expression |
---|---|
caragenelss | ⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragenelss.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
2 | caragenelss.o | . . . . . 6 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
3 | caragenelss.s | . . . . . 6 ⊢ 𝑆 = (CaraGen‘𝑂) | |
4 | 2, 3 | caragenel 41496 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑥 ∩ 𝐴)) +𝑒 (𝑂‘(𝑥 ∖ 𝐴))) = (𝑂‘𝑥)))) |
5 | 1, 4 | mpbid 224 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑥 ∩ 𝐴)) +𝑒 (𝑂‘(𝑥 ∖ 𝐴))) = (𝑂‘𝑥))) |
6 | 5 | simpld 490 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ∪ dom 𝑂) |
7 | caragenelss.x | . . . . . 6 ⊢ 𝑋 = ∪ dom 𝑂 | |
8 | 7 | eqcomi 2834 | . . . . 5 ⊢ ∪ dom 𝑂 = 𝑋 |
9 | 8 | pweqi 4382 | . . . 4 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋 |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋) |
11 | 6, 10 | eleqtrd 2908 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
12 | elpwg 4386 | . . 3 ⊢ (𝐴 ∈ 𝑆 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
13 | 1, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
14 | 11, 13 | mpbid 224 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∀wral 3117 ∖ cdif 3795 ∩ cin 3797 ⊆ wss 3798 𝒫 cpw 4378 ∪ cuni 4658 dom cdm 5342 ‘cfv 6123 (class class class)co 6905 +𝑒 cxad 12230 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-iota 6086 df-fun 6125 df-fv 6131 df-ov 6908 df-caragen 41493 |
This theorem is referenced by: caragenuncllem 41513 caragenuncl 41514 |
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