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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 41647 | . 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 7379 | . . . . . . 7 ⊢ (𝜑 → dom 𝑂 ∈ V) |
7 | uniexg 7234 | . . . . . . 7 ⊢ (dom 𝑂 ∈ V → ∪ dom 𝑂 ∈ V) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) |
9 | 5, 8 | eqeltrd 2859 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ V) |
10 | 9, 4 | ssexd 5044 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
11 | elpwg 4387 | . . . 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 6626 | 1 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ⊆ wss 3792 𝒫 cpw 4379 ∪ cuni 4673 dom cdm 5357 ‘cfv 6137 (class class class)co 6924 0cc0 10274 +∞cpnf 10410 [,]cicc 12494 OutMeascome 41640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-fv 6145 df-ome 41641 |
This theorem is referenced by: caragen0 41657 omexrcl 41658 caragenunidm 41659 omessre 41661 caragenuncllem 41663 caragendifcl 41665 omeunle 41667 omeiunle 41668 omeiunltfirp 41670 carageniuncllem2 41673 carageniuncl 41674 caratheodorylem1 41677 caratheodorylem2 41678 omege0 41684 |
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