Theorem omecl 41511

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 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[,]+∞))

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