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Theorem omeunile 45519
Description: The outer measure of the union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
omeunile.o (𝜑𝑂 ∈ OutMeas)
omeunile.x 𝑋 = dom 𝑂
omeunile.y (𝜑𝑌 ⊆ 𝒫 𝑋)
omeunile.ct (𝜑𝑌 ≼ ω)
Assertion
Ref Expression
omeunile (𝜑 → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))

Proof of Theorem omeunile
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omeunile.ct . 2 (𝜑𝑌 ≼ ω)
2 omeunile.y . . . . 5 (𝜑𝑌 ⊆ 𝒫 𝑋)
3 omeunile.o . . . . . . . . 9 (𝜑𝑂 ∈ OutMeas)
4 omeunile.x . . . . . . . . 9 𝑋 = dom 𝑂
53, 4unidmex 44038 . . . . . . . 8 (𝜑𝑋 ∈ V)
65pwexd 5376 . . . . . . 7 (𝜑 → 𝒫 𝑋 ∈ V)
7 ssexg 5322 . . . . . . 7 ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V)
82, 6, 7syl2anc 582 . . . . . 6 (𝜑𝑌 ∈ V)
9 elpwg 4604 . . . . . 6 (𝑌 ∈ V → (𝑌 ∈ 𝒫 𝒫 𝑋𝑌 ⊆ 𝒫 𝑋))
108, 9syl 17 . . . . 5 (𝜑 → (𝑌 ∈ 𝒫 𝒫 𝑋𝑌 ⊆ 𝒫 𝑋))
112, 10mpbird 256 . . . 4 (𝜑𝑌 ∈ 𝒫 𝒫 𝑋)
12 omedm 45513 . . . . . . 7 (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)
133, 12syl 17 . . . . . 6 (𝜑 → dom 𝑂 = 𝒫 dom 𝑂)
144pweqi 4617 . . . . . . . 8 𝒫 𝑋 = 𝒫 dom 𝑂
1514eqcomi 2739 . . . . . . 7 𝒫 dom 𝑂 = 𝒫 𝑋
1615a1i 11 . . . . . 6 (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝑋)
1713, 16eqtr2d 2771 . . . . 5 (𝜑 → 𝒫 𝑋 = dom 𝑂)
1817pweqd 4618 . . . 4 (𝜑 → 𝒫 𝒫 𝑋 = 𝒫 dom 𝑂)
1911, 18eleqtrd 2833 . . 3 (𝜑𝑌 ∈ 𝒫 dom 𝑂)
20 isome 45508 . . . . . 6 (𝑂 ∈ OutMeas → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑥 ∈ 𝒫 𝑦(𝑂𝑥) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))
213, 20syl 17 . . . . 5 (𝜑 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑥 ∈ 𝒫 𝑦(𝑂𝑥) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))
223, 21mpbid 231 . . . 4 (𝜑 → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑥 ∈ 𝒫 𝑦(𝑂𝑥) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))))
2322simprd 494 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))
24 breq1 5150 . . . . 5 (𝑦 = 𝑌 → (𝑦 ≼ ω ↔ 𝑌 ≼ ω))
25 unieq 4918 . . . . . . 7 (𝑦 = 𝑌 𝑦 = 𝑌)
2625fveq2d 6894 . . . . . 6 (𝑦 = 𝑌 → (𝑂 𝑦) = (𝑂 𝑌))
27 reseq2 5975 . . . . . . 7 (𝑦 = 𝑌 → (𝑂𝑦) = (𝑂𝑌))
2827fveq2d 6894 . . . . . 6 (𝑦 = 𝑌 → (Σ^‘(𝑂𝑦)) = (Σ^‘(𝑂𝑌)))
2926, 28breq12d 5160 . . . . 5 (𝑦 = 𝑌 → ((𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)) ↔ (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌))))
3024, 29imbi12d 343 . . . 4 (𝑦 = 𝑌 → ((𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))) ↔ (𝑌 ≼ ω → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))))
3130rspcva 3609 . . 3 ((𝑌 ∈ 𝒫 dom 𝑂 ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))) → (𝑌 ≼ ω → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌))))
3219, 23, 31syl2anc 582 . 2 (𝜑 → (𝑌 ≼ ω → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌))))
331, 32mpd 15 1 (𝜑 → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  wral 3059  Vcvv 3472  wss 3947  c0 4321  𝒫 cpw 4601   cuni 4907   class class class wbr 5147  dom cdm 5675  cres 5677  wf 6538  cfv 6542  (class class class)co 7411  ωcom 7857  cdom 8939  0cc0 11112  +∞cpnf 11249  cle 11253  [,]cicc 13331  Σ^csumge0 45376  OutMeascome 45503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ome 45504
This theorem is referenced by:  omeunle  45530  omeiunle  45531
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