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 44084 | . 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 7784 | . . . . . . 7 ⊢ (𝜑 → dom 𝑂 ∈ V) |
7 | 6 | uniexd 7627 | . . . . . 6 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) |
8 | 5, 7 | eqeltrd 2837 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ V) |
9 | 8, 4 | ssexd 5257 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
10 | elpwg 4542 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
12 | 4, 11 | mpbird 257 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
13 | 3, 12 | ffvelcdmd 6994 | 1 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2104 Vcvv 3437 ⊆ wss 3892 𝒫 cpw 4539 ∪ cuni 4844 dom cdm 5600 ‘cfv 6458 (class class class)co 7307 0cc0 10917 +∞cpnf 11052 [,]cicc 13128 OutMeascome 44077 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pr 5361 ax-un 7620 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3287 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-fv 6466 df-ome 44078 |
This theorem is referenced by: caragen0 44094 omexrcl 44095 caragenunidm 44096 omessre 44098 caragenuncllem 44100 caragendifcl 44102 omeunle 44104 omeiunle 44105 omeiunltfirp 44107 carageniuncllem2 44110 carageniuncl 44111 caratheodorylem1 44114 caratheodorylem2 44115 omege0 44121 |
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