Theorem omecl 41223

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 41216	. 2 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞))
4		omecl.ss	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
5	2	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑋 = ∪ dom 𝑂)
6		dmexg 7262	. . . . . . . 8 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
7	1, 6	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝑂 ∈ V)
8		uniexg 7120	. . . . . . 7 ⊢ (dom 𝑂 ∈ V → ∪ dom 𝑂 ∈ V)
9	7, 8	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V)
10	5, 9	eqeltrd 2839	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ V)
11	10, 4	ssexd 4957	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
12		elpwg 4310	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
14	4, 13	mpbird 247	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋)
15	3, 14	ffvelrnd 6523	1 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ⊆ wss 3715  𝒫 cpw 4302  ∪ cuni 4588  dom cdm 5266  ‘cfv 6049  (class class class)co 6813  0cc0 10128  +∞cpnf 10263  [,]cicc 12371  OutMeascome 41209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ome 41210

This theorem is referenced by:  caragen0  41226  omexrcl  41227  caragenunidm  41228  omessre  41230  caragenuncllem  41232  caragendifcl  41234  omeunle  41236  omeiunle  41237  omeiunltfirp  41239  carageniuncllem2  41242  carageniuncl  41243  caratheodorylem1  41246  caratheodorylem2  41247  omege0  41253