Theorem omessle 46419

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 44952	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
5		omessle.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑋)
6	4, 5	ssexd 5342	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
7	6, 1	ssexd 5342	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
8		elpwg 4625	. . . 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 4059	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ∪ dom 𝑂)
12		elpwg 4625	. . . . 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 46415	. . . . . 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 4636	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝒫 𝑦 = 𝒫 𝐵)
20	19	raleqdv 3334	. . . . 5 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝑦)))
21		fveq2 6920	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑂‘𝑦) = (𝑂‘𝐵))
22	21	breq2d 5178	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ (𝑂‘𝑧) ≤ (𝑂‘𝐵)))
23	22	ralbidv 3184	. . . . 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 3633	. . 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 583	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵))
27		fveq2 6920	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑂‘𝑧) = (𝑂‘𝐴))
28	27	breq1d 5176	. . 3 ⊢ (𝑧 = 𝐴 → ((𝑂‘𝑧) ≤ (𝑂‘𝐵) ↔ (𝑂‘𝐴) ≤ (𝑂‘𝐵)))
29	28	rspcva 3633	. 2 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵)) → (𝑂‘𝐴) ≤ (𝑂‘𝐵))
30	10, 26, 29	syl2anc 583	1 ⊢ (𝜑 → (𝑂‘𝐴) ≤ (𝑂‘𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  ∪ cuni 4931   class class class wbr 5166  dom cdm 5700   ↾ cres 5702  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ωcom 7903   ≼ cdom 9001  0cc0 11184  +∞cpnf 11321   ≤ cle 11325  [,]cicc 13410  Σ^{^}csumge0 46283  OutMeascome 46410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ome 46411

This theorem is referenced by:  omessre  46431  omeiunltfirp  46440  carageniuncllem2  46443  caratheodorylem2  46448  omess0  46455  caragencmpl  46456