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Theorem omessle 46454
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 (𝜑𝑂 ∈ OutMeas)
omessle.x 𝑋 = dom 𝑂
omessle.b (𝜑𝐵𝑋)
omessle.a (𝜑𝐴𝐵)
Assertion
Ref Expression
omessle (𝜑 → (𝑂𝐴) ≤ (𝑂𝐵))

Proof of Theorem omessle
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omessle.a . . 3 (𝜑𝐴𝐵)
2 omessle.o . . . . . . 7 (𝜑𝑂 ∈ OutMeas)
3 omessle.x . . . . . . 7 𝑋 = dom 𝑂
42, 3unidmex 44990 . . . . . 6 (𝜑𝑋 ∈ V)
5 omessle.b . . . . . 6 (𝜑𝐵𝑋)
64, 5ssexd 5330 . . . . 5 (𝜑𝐵 ∈ V)
76, 1ssexd 5330 . . . 4 (𝜑𝐴 ∈ V)
8 elpwg 4608 . . . 4 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
97, 8syl 17 . . 3 (𝜑 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
101, 9mpbird 257 . 2 (𝜑𝐴 ∈ 𝒫 𝐵)
115, 3sseqtrdi 4046 . . . 4 (𝜑𝐵 dom 𝑂)
12 elpwg 4608 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ 𝒫 dom 𝑂𝐵 dom 𝑂))
136, 12syl 17 . . . 4 (𝜑 → (𝐵 ∈ 𝒫 dom 𝑂𝐵 dom 𝑂))
1411, 13mpbird 257 . . 3 (𝜑𝐵 ∈ 𝒫 dom 𝑂)
15 isome 46450 . . . . . 6 (𝑂 ∈ OutMeas → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))
162, 15syl 17 . . . . 5 (𝜑 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))
172, 16mpbid 232 . . . 4 (𝜑 → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))))
1817simplrd 770 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦))
19 pweq 4619 . . . . . 6 (𝑦 = 𝐵 → 𝒫 𝑦 = 𝒫 𝐵)
2019raleqdv 3324 . . . . 5 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝑦)))
21 fveq2 6907 . . . . . . 7 (𝑦 = 𝐵 → (𝑂𝑦) = (𝑂𝐵))
2221breq2d 5160 . . . . . 6 (𝑦 = 𝐵 → ((𝑂𝑧) ≤ (𝑂𝑦) ↔ (𝑂𝑧) ≤ (𝑂𝐵)))
2322ralbidv 3176 . . . . 5 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵)))
2420, 23bitrd 279 . . . 4 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵)))
2524rspcva 3620 . . 3 ((𝐵 ∈ 𝒫 dom 𝑂 ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) → ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵))
2614, 18, 25syl2anc 584 . 2 (𝜑 → ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵))
27 fveq2 6907 . . . 4 (𝑧 = 𝐴 → (𝑂𝑧) = (𝑂𝐴))
2827breq1d 5158 . . 3 (𝑧 = 𝐴 → ((𝑂𝑧) ≤ (𝑂𝐵) ↔ (𝑂𝐴) ≤ (𝑂𝐵)))
2928rspcva 3620 . 2 ((𝐴 ∈ 𝒫 𝐵 ∧ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵)) → (𝑂𝐴) ≤ (𝑂𝐵))
3010, 26, 29syl2anc 584 1 (𝜑 → (𝑂𝐴) ≤ (𝑂𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912   class class class wbr 5148  dom cdm 5689  cres 5691  wf 6559  cfv 6563  (class class class)co 7431  ωcom 7887  cdom 8982  0cc0 11153  +∞cpnf 11290  cle 11294  [,]cicc 13387  Σ^csumge0 46318  OutMeascome 46445
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ome 46446
This theorem is referenced by:  omessre  46466  omeiunltfirp  46475  carageniuncllem2  46478  caratheodorylem2  46483  omess0  46490  caragencmpl  46491
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