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Theorem omeunile 45221
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 43737 . . . . . . . 8 (𝜑𝑋 ∈ V)
65pwexd 5378 . . . . . . 7 (𝜑 → 𝒫 𝑋 ∈ V)
7 ssexg 5324 . . . . . . 7 ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V)
82, 6, 7syl2anc 585 . . . . . 6 (𝜑𝑌 ∈ V)
9 elpwg 4606 . . . . . 6 (𝑌 ∈ V → (𝑌 ∈ 𝒫 𝒫 𝑋𝑌 ⊆ 𝒫 𝑋))
108, 9syl 17 . . . . 5 (𝜑 → (𝑌 ∈ 𝒫 𝒫 𝑋𝑌 ⊆ 𝒫 𝑋))
112, 10mpbird 257 . . . 4 (𝜑𝑌 ∈ 𝒫 𝒫 𝑋)
12 omedm 45215 . . . . . . 7 (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)
133, 12syl 17 . . . . . 6 (𝜑 → dom 𝑂 = 𝒫 dom 𝑂)
144pweqi 4619 . . . . . . . 8 𝒫 𝑋 = 𝒫 dom 𝑂
1514eqcomi 2742 . . . . . . 7 𝒫 dom 𝑂 = 𝒫 𝑋
1615a1i 11 . . . . . 6 (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝑋)
1713, 16eqtr2d 2774 . . . . 5 (𝜑 → 𝒫 𝑋 = dom 𝑂)
1817pweqd 4620 . . . 4 (𝜑 → 𝒫 𝒫 𝑋 = 𝒫 dom 𝑂)
1911, 18eleqtrd 2836 . . 3 (𝜑𝑌 ∈ 𝒫 dom 𝑂)
20 isome 45210 . . . . . 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 497 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))
24 breq1 5152 . . . . 5 (𝑦 = 𝑌 → (𝑦 ≼ ω ↔ 𝑌 ≼ ω))
25 unieq 4920 . . . . . . 7 (𝑦 = 𝑌 𝑦 = 𝑌)
2625fveq2d 6896 . . . . . 6 (𝑦 = 𝑌 → (𝑂 𝑦) = (𝑂 𝑌))
27 reseq2 5977 . . . . . . 7 (𝑦 = 𝑌 → (𝑂𝑦) = (𝑂𝑌))
2827fveq2d 6896 . . . . . 6 (𝑦 = 𝑌 → (Σ^‘(𝑂𝑦)) = (Σ^‘(𝑂𝑌)))
2926, 28breq12d 5162 . . . . 5 (𝑦 = 𝑌 → ((𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)) ↔ (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌))))
3024, 29imbi12d 345 . . . 4 (𝑦 = 𝑌 → ((𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))) ↔ (𝑌 ≼ ω → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))))
3130rspcva 3611 . . 3 ((𝑌 ∈ 𝒫 dom 𝑂 ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))) → (𝑌 ≼ ω → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌))))
3219, 23, 31syl2anc 585 . 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 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  wss 3949  c0 4323  𝒫 cpw 4603   cuni 4909   class class class wbr 5149  dom cdm 5677  cres 5679  wf 6540  cfv 6544  (class class class)co 7409  ωcom 7855  cdom 8937  0cc0 11110  +∞cpnf 11245  cle 11249  [,]cicc 13327  Σ^csumge0 45078  OutMeascome 45205
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  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ome 45206
This theorem is referenced by:  omeunle  45232  omeiunle  45233
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