Theorem omecl 42792

Description: The outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
omecl.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
omecl.x	⊢ 𝑋 = ∪ dom 𝑂
omecl.ss	⊢ (𝜑 → 𝐴 ⊆ 𝑋)

Assertion

Ref	Expression
omecl	⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞))

Proof of Theorem omecl

Step	Hyp	Ref	Expression
1		omecl.o	. . 3 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
2		omecl.x	. . 3 ⊢ 𝑋 = ∪ dom 𝑂
3	1, 2	omef 42785	. 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 7617	. . . . . . 7 ⊢ (𝜑 → dom 𝑂 ∈ V)
7	6	uniexd 7470	. . . . . 6 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V)
8	5, 7	eqeltrd 2915	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ V)
9	8, 4	ssexd 5230	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
10		elpwg 4544	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
11	9, 10	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
12	4, 11	mpbird 259	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋)
13	3, 12	ffvelrnd 6854	1 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ⊆ wss 3938  𝒫 cpw 4541  ∪ cuni 4840  dom cdm 5557  ‘cfv 6357  (class class class)co 7158  0cc0 10539  +∞cpnf 10674  [,]cicc 12744  OutMeascome 42778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ome 42779

This theorem is referenced by:  caragen0  42795  omexrcl  42796  caragenunidm  42797  omessre  42799  caragenuncllem  42801  caragendifcl  42803  omeunle  42805  omeiunle  42806  omeiunltfirp  42808  carageniuncllem2  42811  carageniuncl  42812  caratheodorylem1  42815  caratheodorylem2  42816  omege0  42822