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Theorem isome 45083
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 5898 . . . . . . 7 (𝑥 = 𝑂 → dom 𝑥 = dom 𝑂)
31, 2feq12d 6695 . . . . . 6 (𝑥 = 𝑂 → (𝑥:dom 𝑥⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞)))
42unieqd 4918 . . . . . . . 8 (𝑥 = 𝑂 dom 𝑥 = dom 𝑂)
54pweqd 4615 . . . . . . 7 (𝑥 = 𝑂 → 𝒫 dom 𝑥 = 𝒫 dom 𝑂)
62, 5eqeq12d 2749 . . . . . 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 6880 . . . . . 6 (𝑥 = 𝑂 → (𝑥‘∅) = (𝑂‘∅))
98eqeq1d 2735 . . . . 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 6880 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑧) = (𝑂𝑧))
12 fveq1 6880 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑦) = (𝑂𝑦))
1311, 12breq12d 5157 . . . . . 6 (𝑥 = 𝑂 → ((𝑥𝑧) ≤ (𝑥𝑦) ↔ (𝑂𝑧) ≤ (𝑂𝑦)))
1413ralbidv 3178 . . . . 5 (𝑥 = 𝑂 → (∀𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)))
155, 14raleqbidv 3343 . . . 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 4615 . . . 4 (𝑥 = 𝑂 → 𝒫 dom 𝑥 = 𝒫 dom 𝑂)
18 fveq1 6880 . . . . . 6 (𝑥 = 𝑂 → (𝑥 𝑦) = (𝑂 𝑦))
19 reseq1 5970 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑦) = (𝑂𝑦))
2019fveq2d 6885 . . . . . 6 (𝑥 = 𝑂 → (Σ^‘(𝑥𝑦)) = (Σ^‘(𝑂𝑦)))
2118, 20breq12d 5157 . . . . 5 (𝑥 = 𝑂 → ((𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦)) ↔ (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))
2221imbi2d 341 . . . 4 (𝑥 = 𝑂 → ((𝑦 ≼ ω → (𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦))) ↔ (𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))))
2317, 22raleqbidv 3343 . . 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 45079 . 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 3668 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 205  wa 397   = wceq 1542  wcel 2107  wral 3062  c0 4320  𝒫 cpw 4598   cuni 4904   class class class wbr 5144  dom cdm 5672  cres 5674  wf 6531  cfv 6535  (class class class)co 7396  ωcom 7842  cdom 8925  0cc0 11097  +∞cpnf 11232  cle 11236  [,]cicc 13314  Σ^csumge0 44951  OutMeascome 45078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-fv 6543  df-ome 45079
This theorem is referenced by:  omef  45085  ome0  45086  omessle  45087  omedm  45088  omeunile  45094  0ome  45118  isomennd  45120
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