Theorem omessle 47097

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 45655	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
5		omessle.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑋)
6	4, 5	ssexd 5292	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
7	6, 1	ssexd 5292	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
8		elpwg 4567	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
9	7, 8	syl 18	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
10	1, 9	mpbird 260	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)
11	5, 3	sseqtrdi 3985	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ∪ dom 𝑂)
12		elpwg 4567	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝒫 ∪ dom 𝑂 ↔ 𝐵 ⊆ ∪ dom 𝑂))
13	6, 12	syl 18	. . . 4 ⊢ (𝜑 → (𝐵 ∈ 𝒫 ∪ dom 𝑂 ↔ 𝐵 ⊆ ∪ dom 𝑂))
14	11, 13	mpbird 260	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝒫 ∪ dom 𝑂)
15		isome 47093	. . . . . 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 18	. . . . 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 235	. . . 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 781	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦))
19		pweq 4578	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝒫 𝑦 = 𝒫 𝐵)
20	19	raleqdv 3329	. . . . 5 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝑦)))
21		fveq2 6879	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑂‘𝑦) = (𝑂‘𝐵))
22	21	breq2d 5122	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ (𝑂‘𝑧) ≤ (𝑂‘𝐵)))
23	22	ralbidv 3194	. . . . 5 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵)))
24	20, 23	bitrd 282	. . . 4 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵)))
25	24	rspcva 3588	. . 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 595	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵))
27		fveq2 6879	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑂‘𝑧) = (𝑂‘𝐴))
28	27	breq1d 5120	. . 3 ⊢ (𝑧 = 𝐴 → ((𝑂‘𝑧) ≤ (𝑂‘𝐵) ↔ (𝑂‘𝐴) ≤ (𝑂‘𝐵)))
29	28	rspcva 3588	. 2 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ∀𝑧 ∈ 𝒫 𝐵(𝑂‘𝑧) ≤ (𝑂‘𝐵)) → (𝑂‘𝐴) ≤ (𝑂‘𝐵))
30	10, 26, 29	syl2anc 595	1 ⊢ (𝜑 → (𝑂‘𝐴) ≤ (𝑂‘𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4564  ∪ cuni 4873   class class class wbr 5110  dom cdm 5659   ↾ cres 5661  ⟶wf 6529  ‘cfv 6533  (class class class)co 7408  ωcom 7858   ≼ cdom 8937  0cc0 11096  +∞cpnf 11236   ≤ cle 11240  [,]cicc 13371  Σ^{^}csumge0 46961  OutMeascome 47088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ome 47089

This theorem is referenced by:  omessre  47109  omeiunltfirp  47118  carageniuncllem2  47121  caratheodorylem2  47126  omess0  47133  caragencmpl  47134