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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omecl | Structured version Visualization version GIF version |
Description: The outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
omecl.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
omecl.x | ⊢ 𝑋 = ∪ dom 𝑂 |
omecl.ss | ⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
Ref | Expression |
---|---|
omecl | ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omecl.o | . . 3 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
2 | omecl.x | . . 3 ⊢ 𝑋 = ∪ dom 𝑂 | |
3 | 1, 2 | omef 41504 | . 2 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) |
4 | omecl.ss | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | |
5 | 2 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑋 = ∪ dom 𝑂) |
6 | 1 | dmexd 7360 | . . . . . . 7 ⊢ (𝜑 → dom 𝑂 ∈ V) |
7 | uniexg 7215 | . . . . . . 7 ⊢ (dom 𝑂 ∈ V → ∪ dom 𝑂 ∈ V) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) |
9 | 5, 8 | eqeltrd 2906 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ V) |
10 | 9, 4 | ssexd 5030 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
11 | elpwg 4386 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
13 | 4, 12 | mpbird 249 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
14 | 3, 13 | ffvelrnd 6609 | 1 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1658 ∈ wcel 2166 Vcvv 3414 ⊆ wss 3798 𝒫 cpw 4378 ∪ cuni 4658 dom cdm 5342 ‘cfv 6123 (class class class)co 6905 0cc0 10252 +∞cpnf 10388 [,]cicc 12466 OutMeascome 41497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 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-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-fv 6131 df-ome 41498 |
This theorem is referenced by: caragen0 41514 omexrcl 41515 caragenunidm 41516 omessre 41518 caragenuncllem 41520 caragendifcl 41522 omeunle 41524 omeiunle 41525 omeiunltfirp 41527 carageniuncllem2 41530 carageniuncl 41531 caratheodorylem1 41534 caratheodorylem2 41535 omege0 41541 |
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