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Theorem isome 42999
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 5760 . . . . . . 7 (𝑥 = 𝑂 → dom 𝑥 = dom 𝑂)
31, 2feq12d 6491 . . . . . 6 (𝑥 = 𝑂 → (𝑥:dom 𝑥⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞)))
42unieqd 4839 . . . . . . . 8 (𝑥 = 𝑂 dom 𝑥 = dom 𝑂)
54pweqd 4541 . . . . . . 7 (𝑥 = 𝑂 → 𝒫 dom 𝑥 = 𝒫 dom 𝑂)
62, 5eqeq12d 2840 . . . . . 6 (𝑥 = 𝑂 → (dom 𝑥 = 𝒫 dom 𝑥 ↔ dom 𝑂 = 𝒫 dom 𝑂))
73, 6anbi12d 633 . . . . 5 (𝑥 = 𝑂 → ((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 dom 𝑥) ↔ (𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂)))
8 fveq1 6658 . . . . . 6 (𝑥 = 𝑂 → (𝑥‘∅) = (𝑂‘∅))
98eqeq1d 2826 . . . . 5 (𝑥 = 𝑂 → ((𝑥‘∅) = 0 ↔ (𝑂‘∅) = 0))
107, 9anbi12d 633 . . . 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 6658 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑧) = (𝑂𝑧))
12 fveq1 6658 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑦) = (𝑂𝑦))
1311, 12breq12d 5066 . . . . . 6 (𝑥 = 𝑂 → ((𝑥𝑧) ≤ (𝑥𝑦) ↔ (𝑂𝑧) ≤ (𝑂𝑦)))
1413ralbidv 3192 . . . . 5 (𝑥 = 𝑂 → (∀𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)))
155, 14raleqbidv 3393 . . . 4 (𝑥 = 𝑂 → (∀𝑦 ∈ 𝒫 dom 𝑥𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦) ↔ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)))
1610, 15anbi12d 633 . . 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 4541 . . . 4 (𝑥 = 𝑂 → 𝒫 dom 𝑥 = 𝒫 dom 𝑂)
18 fveq1 6658 . . . . . 6 (𝑥 = 𝑂 → (𝑥 𝑦) = (𝑂 𝑦))
19 reseq1 5835 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑦) = (𝑂𝑦))
2019fveq2d 6663 . . . . . 6 (𝑥 = 𝑂 → (Σ^‘(𝑥𝑦)) = (Σ^‘(𝑂𝑦)))
2118, 20breq12d 5066 . . . . 5 (𝑥 = 𝑂 → ((𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦)) ↔ (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))
2221imbi2d 344 . . . 4 (𝑥 = 𝑂 → ((𝑦 ≼ ω → (𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦))) ↔ (𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))))
2317, 22raleqbidv 3393 . . 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 633 . 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 42995 . 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 3654 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 209  wa 399   = wceq 1538  wcel 2115  wral 3133  c0 4276  𝒫 cpw 4522   cuni 4825   class class class wbr 5053  dom cdm 5543  cres 5545  wf 6340  cfv 6344  (class class class)co 7146  ωcom 7571  cdom 8499  0cc0 10531  +∞cpnf 10666  cle 10670  [,]cicc 12736  Σ^csumge0 42867  OutMeascome 42994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352  df-ome 42995
This theorem is referenced by:  omef  43001  ome0  43002  omessle  43003  omedm  43004  omeunile  43010  0ome  43034  isomennd  43036
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