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Theorem isome 40471
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 5313 . . . . . . 7 (𝑥 = 𝑂 → dom 𝑥 = dom 𝑂)
31, 2feq12d 6020 . . . . . 6 (𝑥 = 𝑂 → (𝑥:dom 𝑥⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞)))
42unieqd 4437 . . . . . . . 8 (𝑥 = 𝑂 dom 𝑥 = dom 𝑂)
54pweqd 4154 . . . . . . 7 (𝑥 = 𝑂 → 𝒫 dom 𝑥 = 𝒫 dom 𝑂)
62, 5eqeq12d 2635 . . . . . 6 (𝑥 = 𝑂 → (dom 𝑥 = 𝒫 dom 𝑥 ↔ dom 𝑂 = 𝒫 dom 𝑂))
73, 6anbi12d 746 . . . . 5 (𝑥 = 𝑂 → ((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 dom 𝑥) ↔ (𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂)))
8 fveq1 6177 . . . . . 6 (𝑥 = 𝑂 → (𝑥‘∅) = (𝑂‘∅))
98eqeq1d 2622 . . . . 5 (𝑥 = 𝑂 → ((𝑥‘∅) = 0 ↔ (𝑂‘∅) = 0))
107, 9anbi12d 746 . . . 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 6177 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑧) = (𝑂𝑧))
12 fveq1 6177 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑦) = (𝑂𝑦))
1311, 12breq12d 4657 . . . . . 6 (𝑥 = 𝑂 → ((𝑥𝑧) ≤ (𝑥𝑦) ↔ (𝑂𝑧) ≤ (𝑂𝑦)))
1413ralbidv 2983 . . . . 5 (𝑥 = 𝑂 → (∀𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)))
155, 14raleqbidv 3147 . . . 4 (𝑥 = 𝑂 → (∀𝑦 ∈ 𝒫 dom 𝑥𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦) ↔ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)))
1610, 15anbi12d 746 . . 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 4154 . . . 4 (𝑥 = 𝑂 → 𝒫 dom 𝑥 = 𝒫 dom 𝑂)
18 fveq1 6177 . . . . . 6 (𝑥 = 𝑂 → (𝑥 𝑦) = (𝑂 𝑦))
19 reseq1 5379 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑦) = (𝑂𝑦))
2019fveq2d 6182 . . . . . 6 (𝑥 = 𝑂 → (Σ^‘(𝑥𝑦)) = (Σ^‘(𝑂𝑦)))
2118, 20breq12d 4657 . . . . 5 (𝑥 = 𝑂 → ((𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦)) ↔ (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))
2221imbi2d 330 . . . 4 (𝑥 = 𝑂 → ((𝑦 ≼ ω → (𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦))) ↔ (𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))))
2317, 22raleqbidv 3147 . . 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 746 . 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 40467 . 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 3347 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 196  wa 384   = wceq 1481  wcel 1988  wral 2909  c0 3907  𝒫 cpw 4149   cuni 4427   class class class wbr 4644  dom cdm 5104  cres 5106  wf 5872  cfv 5876  (class class class)co 6635  ωcom 7050  cdom 7938  0cc0 9921  +∞cpnf 10056  cle 10060  [,]cicc 12163  Σ^csumge0 40342  OutMeascome 40466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ome 40467
This theorem is referenced by:  omef  40473  ome0  40474  omessle  40475  omedm  40476  omeunile  40482  0ome  40506  isomennd  40508
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