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Theorem isome 46450
Description: Express the predicate "𝑂 is an outer measure." Definition 113A of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
isome (𝑂𝑉 → (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))))))
Distinct variable group:   𝑦,𝑂,𝑧
Allowed substitution hints:   𝑉(𝑦,𝑧)

Proof of Theorem isome
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . . 7 (𝑥 = 𝑂𝑥 = 𝑂)
2 dmeq 5917 . . . . . . 7 (𝑥 = 𝑂 → dom 𝑥 = dom 𝑂)
31, 2feq12d 6725 . . . . . 6 (𝑥 = 𝑂 → (𝑥:dom 𝑥⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞)))
42unieqd 4925 . . . . . . . 8 (𝑥 = 𝑂 dom 𝑥 = dom 𝑂)
54pweqd 4622 . . . . . . 7 (𝑥 = 𝑂 → 𝒫 dom 𝑥 = 𝒫 dom 𝑂)
62, 5eqeq12d 2751 . . . . . 6 (𝑥 = 𝑂 → (dom 𝑥 = 𝒫 dom 𝑥 ↔ dom 𝑂 = 𝒫 dom 𝑂))
73, 6anbi12d 632 . . . . 5 (𝑥 = 𝑂 → ((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 dom 𝑥) ↔ (𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂)))
8 fveq1 6906 . . . . . 6 (𝑥 = 𝑂 → (𝑥‘∅) = (𝑂‘∅))
98eqeq1d 2737 . . . . 5 (𝑥 = 𝑂 → ((𝑥‘∅) = 0 ↔ (𝑂‘∅) = 0))
107, 9anbi12d 632 . . . 4 (𝑥 = 𝑂 → (((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 dom 𝑥) ∧ (𝑥‘∅) = 0) ↔ ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0)))
11 fveq1 6906 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑧) = (𝑂𝑧))
12 fveq1 6906 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑦) = (𝑂𝑦))
1311, 12breq12d 5161 . . . . . 6 (𝑥 = 𝑂 → ((𝑥𝑧) ≤ (𝑥𝑦) ↔ (𝑂𝑧) ≤ (𝑂𝑦)))
1413ralbidv 3176 . . . . 5 (𝑥 = 𝑂 → (∀𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)))
155, 14raleqbidv 3344 . . . 4 (𝑥 = 𝑂 → (∀𝑦 ∈ 𝒫 dom 𝑥𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦) ↔ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)))
1610, 15anbi12d 632 . . 3 (𝑥 = 𝑂 → ((((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 dom 𝑥) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦)) ↔ (((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦))))
172pweqd 4622 . . . 4 (𝑥 = 𝑂 → 𝒫 dom 𝑥 = 𝒫 dom 𝑂)
18 fveq1 6906 . . . . . 6 (𝑥 = 𝑂 → (𝑥 𝑦) = (𝑂 𝑦))
19 reseq1 5994 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑦) = (𝑂𝑦))
2019fveq2d 6911 . . . . . 6 (𝑥 = 𝑂 → (Σ^‘(𝑥𝑦)) = (Σ^‘(𝑂𝑦)))
2118, 20breq12d 5161 . . . . 5 (𝑥 = 𝑂 → ((𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦)) ↔ (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))
2221imbi2d 340 . . . 4 (𝑥 = 𝑂 → ((𝑦 ≼ ω → (𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦))) ↔ (𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))))
2317, 22raleqbidv 3344 . . 3 (𝑥 = 𝑂 → (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦))) ↔ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))))
2416, 23anbi12d 632 . 2 (𝑥 = 𝑂 → (((((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 dom 𝑥) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦)))) ↔ ((((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))))))
25 df-ome 46446 . 2 OutMeas = {𝑥 ∣ ((((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 dom 𝑥) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦))))}
2624, 25elab2g 3683 1 (𝑂𝑉 → (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))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  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
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-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:  omef  46452  ome0  46453  omessle  46454  omedm  46455  omeunile  46461  0ome  46485  isomennd  46487
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