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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 46451 | . . . . 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 232 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑥 ∩ 𝐴)) +𝑒 (𝑂‘(𝑥 ∖ 𝐴))) = (𝑂‘𝑥))) |
6 | 5 | simpld 494 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ∪ dom 𝑂) |
7 | caragenelss.x | . . . . . 6 ⊢ 𝑋 = ∪ dom 𝑂 | |
8 | 7 | eqcomi 2744 | . . . . 5 ⊢ ∪ dom 𝑂 = 𝑋 |
9 | 8 | pweqi 4621 | . . . 4 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋 |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋) |
11 | 6, 10 | eleqtrd 2841 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
12 | elpwg 4608 | . . 3 ⊢ (𝐴 ∈ 𝑆 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
13 | 1, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
14 | 11, 13 | mpbid 232 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∖ cdif 3960 ∩ cin 3962 ⊆ wss 3963 𝒫 cpw 4605 ∪ cuni 4912 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 +𝑒 cxad 13150 OutMeascome 46445 CaraGenccaragen 46447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-caragen 46448 |
This theorem is referenced by: caragenuncllem 46468 caragenuncl 46469 |
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