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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 46510 | . . . . 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 2746 | . . . . 5 ⊢ ∪ dom 𝑂 = 𝑋 |
| 9 | 8 | pweqi 4616 | . . . 4 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋 |
| 10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋) |
| 11 | 6, 10 | eleqtrd 2843 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
| 12 | elpwg 4603 | . . 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 1540 ∈ wcel 2108 ∀wral 3061 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 +𝑒 cxad 13152 OutMeascome 46504 CaraGenccaragen 46506 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-caragen 46507 |
| This theorem is referenced by: caragenuncllem 46527 caragenuncl 46528 |
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