Theorem omessle 42773

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 41305	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
5		omessle.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑋)
6	4, 5	ssexd 5221	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
7	6, 1	ssexd 5221	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
8		elpwg 4545	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
10	1, 9	mpbird 259	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)
11	5, 3	sseqtrdi 4017	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ∪ dom 𝑂)
12		elpwg 4545	. . . . 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 259	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝒫 ∪ dom 𝑂)
15		isome 42769	. . . . . 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 234	. . . 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 768	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦))
19		pweq 4542	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝒫 𝑦 = 𝒫 𝐵)
20	19	raleqdv 3416	. . . . 5 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝑦)))
21		fveq2 6665	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑂‘𝑦) = (𝑂‘𝐵))
22	21	breq2d 5071	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ (𝑂‘𝑧) ≤ (𝑂‘𝐵)))
23	22	ralbidv 3197	. . . . 5 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵)))
24	20, 23	bitrd 281	. . . 4 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵)))
25	24	rspcva 3621	. . 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 586	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵))
27		fveq2 6665	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑂‘𝑧) = (𝑂‘𝐴))
28	27	breq1d 5069	. . 3 ⊢ (𝑧 = 𝐴 → ((𝑂‘𝑧) ≤ (𝑂‘𝐵) ↔ (𝑂‘𝐴) ≤ (𝑂‘𝐵)))
29	28	rspcva 3621	. 2 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵)) → (𝑂‘𝐴) ≤ (𝑂‘𝐵))
30	10, 26, 29	syl2anc 586	1 ⊢ (𝜑 → (𝑂‘𝐴) ≤ (𝑂‘𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3495   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  ∪ cuni 4832   class class class wbr 5059  dom cdm 5550   ↾ cres 5552  ⟶wf 6346  ‘cfv 6350  (class class class)co 7150  ωcom 7574   ≼ cdom 8501  0cc0 10531  +∞cpnf 10666   ≤ cle 10670  [,]cicc 12735  Σ^{^}csumge0 42637  OutMeascome 42764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-ome 42765

This theorem is referenced by:  omessre  42785  omeiunltfirp  42794  carageniuncllem2  42797  caratheodorylem2  42802  omess0  42809  caragencmpl  42810