Theorem omessle 46623

Description: The outer measure of a set is greater than or equal to the measure of a subset, Definition 113A (ii) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
omessle.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
omessle.x	⊢ 𝑋 = ∪ dom 𝑂
omessle.b	⊢ (𝜑 → 𝐵 ⊆ 𝑋)
omessle.a	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
omessle	⊢ (𝜑 → (𝑂‘𝐴) ≤ (𝑂‘𝐵))

Proof of Theorem omessle

Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omessle.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		omessle.o	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
3		omessle.x	. . . . . . 7 ⊢ 𝑋 = ∪ dom 𝑂
4	2, 3	unidmex 45174	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
5		omessle.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑋)
6	4, 5	ssexd 5266	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
7	6, 1	ssexd 5266	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
8		elpwg 4554	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
10	1, 9	mpbird 257	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)
11	5, 3	sseqtrdi 3971	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ∪ dom 𝑂)
12		elpwg 4554	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝒫 ∪ dom 𝑂 ↔ 𝐵 ⊆ ∪ dom 𝑂))
13	6, 12	syl 17	. . . 4 ⊢ (𝜑 → (𝐵 ∈ 𝒫 ∪ dom 𝑂 ↔ 𝐵 ⊆ ∪ dom 𝑂))
14	11, 13	mpbird 257	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝒫 ∪ dom 𝑂)
15		isome 46619	. . . . . 6 ⊢ (𝑂 ∈ OutMeas → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))))
16	2, 15	syl 17	. . . . 5 ⊢ (𝜑 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))))
17	2, 16	mpbid 232	. . . 4 ⊢ (𝜑 → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))))
18	17	simplrd 769	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦))
19		pweq 4565	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝒫 𝑦 = 𝒫 𝐵)
20	19	raleqdv 3293	. . . . 5 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝑦)))
21		fveq2 6830	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑂‘𝑦) = (𝑂‘𝐵))
22	21	breq2d 5107	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ (𝑂‘𝑧) ≤ (𝑂‘𝐵)))
23	22	ralbidv 3156	. . . . 5 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵)))
24	20, 23	bitrd 279	. . . 4 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵)))
25	24	rspcva 3571	. . 3 ⊢ ((𝐵 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) → ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵))
26	14, 18, 25	syl2anc 584	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵))
27		fveq2 6830	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑂‘𝑧) = (𝑂‘𝐴))
28	27	breq1d 5105	. . 3 ⊢ (𝑧 = 𝐴 → ((𝑂‘𝑧) ≤ (𝑂‘𝐵) ↔ (𝑂‘𝐴) ≤ (𝑂‘𝐵)))
29	28	rspcva 3571	. 2 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵)) → (𝑂‘𝐴) ≤ (𝑂‘𝐵))
30	10, 26, 29	syl2anc 584	1 ⊢ (𝜑 → (𝑂‘𝐴) ≤ (𝑂‘𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  Vcvv 3437   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4551  ∪ cuni 4860   class class class wbr 5095  dom cdm 5621   ↾ cres 5623  ⟶wf 6484  ‘cfv 6488  (class class class)co 7354  ωcom 7804   ≼ cdom 8875  0cc0 11015  +∞cpnf 11152   ≤ cle 11156  [,]cicc 13252  Σ^{^}csumge0 46487  OutMeascome 46614

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fv 6496  df-ome 46615

This theorem is referenced by:  omessre  46635  omeiunltfirp  46644  carageniuncllem2  46647  caratheodorylem2  46652  omess0  46659  caragencmpl  46660