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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 42770 | . . . . 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 234 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑥 ∩ 𝐴)) +𝑒 (𝑂‘(𝑥 ∖ 𝐴))) = (𝑂‘𝑥))) |
6 | 5 | simpld 497 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ∪ dom 𝑂) |
7 | caragenelss.x | . . . . . 6 ⊢ 𝑋 = ∪ dom 𝑂 | |
8 | 7 | eqcomi 2830 | . . . . 5 ⊢ ∪ dom 𝑂 = 𝑋 |
9 | 8 | pweqi 4543 | . . . 4 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋 |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋) |
11 | 6, 10 | eleqtrd 2915 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
12 | elpwg 4545 | . . 3 ⊢ (𝐴 ∈ 𝑆 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
13 | 1, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
14 | 11, 13 | mpbid 234 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4832 dom cdm 5550 ‘cfv 6350 (class class class)co 7150 +𝑒 cxad 12499 OutMeascome 42764 CaraGenccaragen 42766 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fv 6358 df-ov 7153 df-caragen 42767 |
This theorem is referenced by: caragenuncllem 42787 caragenuncl 42788 |
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