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Theorem omeunile 41040
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 39531 . . . . . . . 8 (𝜑𝑋 ∈ V)
6 pwexg 4880 . . . . . . . 8 (𝑋 ∈ V → 𝒫 𝑋 ∈ V)
75, 6syl 17 . . . . . . 7 (𝜑 → 𝒫 𝑋 ∈ V)
8 ssexg 4837 . . . . . . 7 ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V)
92, 7, 8syl2anc 694 . . . . . 6 (𝜑𝑌 ∈ V)
10 elpwg 4199 . . . . . 6 (𝑌 ∈ V → (𝑌 ∈ 𝒫 𝒫 𝑋𝑌 ⊆ 𝒫 𝑋))
119, 10syl 17 . . . . 5 (𝜑 → (𝑌 ∈ 𝒫 𝒫 𝑋𝑌 ⊆ 𝒫 𝑋))
122, 11mpbird 247 . . . 4 (𝜑𝑌 ∈ 𝒫 𝒫 𝑋)
13 omedm 41034 . . . . . . 7 (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)
143, 13syl 17 . . . . . 6 (𝜑 → dom 𝑂 = 𝒫 dom 𝑂)
154pweqi 4195 . . . . . . . 8 𝒫 𝑋 = 𝒫 dom 𝑂
1615eqcomi 2660 . . . . . . 7 𝒫 dom 𝑂 = 𝒫 𝑋
1716a1i 11 . . . . . 6 (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝑋)
1814, 17eqtr2d 2686 . . . . 5 (𝜑 → 𝒫 𝑋 = dom 𝑂)
1918pweqd 4196 . . . 4 (𝜑 → 𝒫 𝒫 𝑋 = 𝒫 dom 𝑂)
2012, 19eleqtrd 2732 . . 3 (𝜑𝑌 ∈ 𝒫 dom 𝑂)
21 isome 41029 . . . . . 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))))))
223, 21syl 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))))))
233, 22mpbid 222 . . . 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)))))
2423simprd 478 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))
25 breq1 4688 . . . . 5 (𝑦 = 𝑌 → (𝑦 ≼ ω ↔ 𝑌 ≼ ω))
26 unieq 4476 . . . . . . 7 (𝑦 = 𝑌 𝑦 = 𝑌)
2726fveq2d 6233 . . . . . 6 (𝑦 = 𝑌 → (𝑂 𝑦) = (𝑂 𝑌))
28 reseq2 5423 . . . . . . 7 (𝑦 = 𝑌 → (𝑂𝑦) = (𝑂𝑌))
2928fveq2d 6233 . . . . . 6 (𝑦 = 𝑌 → (Σ^‘(𝑂𝑦)) = (Σ^‘(𝑂𝑌)))
3027, 29breq12d 4698 . . . . 5 (𝑦 = 𝑌 → ((𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)) ↔ (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌))))
3125, 30imbi12d 333 . . . 4 (𝑦 = 𝑌 → ((𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))) ↔ (𝑌 ≼ ω → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))))
3231rspcva 3338 . . 3 ((𝑌 ∈ 𝒫 dom 𝑂 ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))) → (𝑌 ≼ ω → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌))))
3320, 24, 32syl2anc 694 . 2 (𝜑 → (𝑌 ≼ ω → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌))))
341, 33mpd 15 1 (𝜑 → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468   class class class wbr 4685  dom cdm 5143  cres 5145  wf 5922  cfv 5926  (class class class)co 6690  ωcom 7107  cdom 7995  0cc0 9974  +∞cpnf 10109  cle 10113  [,]cicc 12216  Σ^csumge0 40897  OutMeascome 41024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ome 41025
This theorem is referenced by:  omeunle  41051  omeiunle  41052
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