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Theorem omessle 39388
Description: The outer measure of a set is larger 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 38242 . . . . . 6 (𝜑𝑋 ∈ V)
5 omessle.b . . . . . 6 (𝜑𝐵𝑋)
64, 5ssexd 4733 . . . . 5 (𝜑𝐵 ∈ V)
76, 1ssexd 4733 . . . 4 (𝜑𝐴 ∈ V)
8 elpwg 4116 . . . 4 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
97, 8syl 17 . . 3 (𝜑 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
101, 9mpbird 246 . 2 (𝜑𝐴 ∈ 𝒫 𝐵)
115, 3syl6sseq 3614 . . . 4 (𝜑𝐵 dom 𝑂)
12 elpwg 4116 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ 𝒫 dom 𝑂𝐵 dom 𝑂))
136, 12syl 17 . . . 4 (𝜑 → (𝐵 ∈ 𝒫 dom 𝑂𝐵 dom 𝑂))
1411, 13mpbird 246 . . 3 (𝜑𝐵 ∈ 𝒫 dom 𝑂)
15 isome 39384 . . . . . 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 221 . . . 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 789 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦))
19 pweq 4111 . . . . . 6 (𝑦 = 𝐵 → 𝒫 𝑦 = 𝒫 𝐵)
2019raleqdv 3121 . . . . 5 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝑦)))
21 fveq2 6103 . . . . . . 7 (𝑦 = 𝐵 → (𝑂𝑦) = (𝑂𝐵))
2221breq2d 4595 . . . . . 6 (𝑦 = 𝐵 → ((𝑂𝑧) ≤ (𝑂𝑦) ↔ (𝑂𝑧) ≤ (𝑂𝐵)))
2322ralbidv 2969 . . . . 5 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵)))
2420, 23bitrd 267 . . . 4 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵)))
2524rspcva 3280 . . 3 ((𝐵 ∈ 𝒫 dom 𝑂 ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) → ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵))
2614, 18, 25syl2anc 691 . 2 (𝜑 → ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵))
27 fveq2 6103 . . . 4 (𝑧 = 𝐴 → (𝑂𝑧) = (𝑂𝐴))
2827breq1d 4593 . . 3 (𝑧 = 𝐴 → ((𝑂𝑧) ≤ (𝑂𝐵) ↔ (𝑂𝐴) ≤ (𝑂𝐵)))
2928rspcva 3280 . 2 ((𝐴 ∈ 𝒫 𝐵 ∧ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵)) → (𝑂𝐴) ≤ (𝑂𝐵))
3010, 26, 29syl2anc 691 1 (𝜑 → (𝑂𝐴) ≤ (𝑂𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  wss 3540  c0 3874  𝒫 cpw 4108   cuni 4372   class class class wbr 4583  dom cdm 5038  cres 5040  wf 5800  cfv 5804  (class class class)co 6549  ωcom 6957  cdom 7839  0cc0 9815  +∞cpnf 9950  cle 9954  [,]cicc 12049  Σ^csumge0 39255  OutMeascome 39379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ome 39380
This theorem is referenced by:  omessre  39400  omeiunltfirp  39409  carageniuncllem2  39412  caratheodorylem2  39417  omess0  39424  caragencmpl  39425
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