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Theorem omessle 42657
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 41189 . . . . . 6 (𝜑𝑋 ∈ V)
5 omessle.b . . . . . 6 (𝜑𝐵𝑋)
64, 5ssexd 5219 . . . . 5 (𝜑𝐵 ∈ V)
76, 1ssexd 5219 . . . 4 (𝜑𝐴 ∈ V)
8 elpwg 4541 . . . 4 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
97, 8syl 17 . . 3 (𝜑 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
101, 9mpbird 258 . 2 (𝜑𝐴 ∈ 𝒫 𝐵)
115, 3sseqtrdi 4014 . . . 4 (𝜑𝐵 dom 𝑂)
12 elpwg 4541 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ 𝒫 dom 𝑂𝐵 dom 𝑂))
136, 12syl 17 . . . 4 (𝜑 → (𝐵 ∈ 𝒫 dom 𝑂𝐵 dom 𝑂))
1411, 13mpbird 258 . . 3 (𝜑𝐵 ∈ 𝒫 dom 𝑂)
15 isome 42653 . . . . . 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 233 . . . 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 766 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦))
19 pweq 4538 . . . . . 6 (𝑦 = 𝐵 → 𝒫 𝑦 = 𝒫 𝐵)
2019raleqdv 3413 . . . . 5 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝑦)))
21 fveq2 6663 . . . . . . 7 (𝑦 = 𝐵 → (𝑂𝑦) = (𝑂𝐵))
2221breq2d 5069 . . . . . 6 (𝑦 = 𝐵 → ((𝑂𝑧) ≤ (𝑂𝑦) ↔ (𝑂𝑧) ≤ (𝑂𝐵)))
2322ralbidv 3194 . . . . 5 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵)))
2420, 23bitrd 280 . . . 4 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵)))
2524rspcva 3618 . . 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 6663 . . . 4 (𝑧 = 𝐴 → (𝑂𝑧) = (𝑂𝐴))
2827breq1d 5067 . . 3 (𝑧 = 𝐴 → ((𝑂𝑧) ≤ (𝑂𝐵) ↔ (𝑂𝐴) ≤ (𝑂𝐵)))
2928rspcva 3618 . 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 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  wss 3933  c0 4288  𝒫 cpw 4535   cuni 4830   class class class wbr 5057  dom cdm 5548  cres 5550  wf 6344  cfv 6348  (class class class)co 7145  ωcom 7569  cdom 8495  0cc0 10525  +∞cpnf 10660  cle 10664  [,]cicc 12729  Σ^csumge0 42521  OutMeascome 42648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ome 42649
This theorem is referenced by:  omessre  42669  omeiunltfirp  42678  carageniuncllem2  42681  caratheodorylem2  42686  omess0  42693  caragencmpl  42694
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