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Theorem omeunile 41245
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 39750 . . . . . . . 8 (𝜑𝑋 ∈ V)
6 pwexg 4994 . . . . . . . 8 (𝑋 ∈ V → 𝒫 𝑋 ∈ V)
75, 6syl 17 . . . . . . 7 (𝜑 → 𝒫 𝑋 ∈ V)
8 ssexg 4952 . . . . . . 7 ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V)
92, 7, 8syl2anc 574 . . . . . 6 (𝜑𝑌 ∈ V)
10 elpwg 4315 . . . . . 6 (𝑌 ∈ V → (𝑌 ∈ 𝒫 𝒫 𝑋𝑌 ⊆ 𝒫 𝑋))
119, 10syl 17 . . . . 5 (𝜑 → (𝑌 ∈ 𝒫 𝒫 𝑋𝑌 ⊆ 𝒫 𝑋))
122, 11mpbird 248 . . . 4 (𝜑𝑌 ∈ 𝒫 𝒫 𝑋)
13 omedm 41239 . . . . . . 7 (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)
143, 13syl 17 . . . . . 6 (𝜑 → dom 𝑂 = 𝒫 dom 𝑂)
154pweqi 4311 . . . . . . . 8 𝒫 𝑋 = 𝒫 dom 𝑂
1615eqcomi 2783 . . . . . . 7 𝒫 dom 𝑂 = 𝒫 𝑋
1716a1i 11 . . . . . 6 (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝑋)
1814, 17eqtr2d 2809 . . . . 5 (𝜑 → 𝒫 𝑋 = dom 𝑂)
1918pweqd 4312 . . . 4 (𝜑 → 𝒫 𝒫 𝑋 = 𝒫 dom 𝑂)
2012, 19eleqtrd 2855 . . 3 (𝜑𝑌 ∈ 𝒫 dom 𝑂)
21 isome 41234 . . . . . 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 223 . . . 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 484 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))
25 breq1 4800 . . . . 5 (𝑦 = 𝑌 → (𝑦 ≼ ω ↔ 𝑌 ≼ ω))
26 unieq 4593 . . . . . . 7 (𝑦 = 𝑌 𝑦 = 𝑌)
2726fveq2d 6352 . . . . . 6 (𝑦 = 𝑌 → (𝑂 𝑦) = (𝑂 𝑌))
28 reseq2 5541 . . . . . . 7 (𝑦 = 𝑌 → (𝑂𝑦) = (𝑂𝑌))
2928fveq2d 6352 . . . . . 6 (𝑦 = 𝑌 → (Σ^‘(𝑂𝑦)) = (Σ^‘(𝑂𝑌)))
3027, 29breq12d 4810 . . . . 5 (𝑦 = 𝑌 → ((𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)) ↔ (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌))))
3125, 30imbi12d 334 . . . 4 (𝑦 = 𝑌 → ((𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))) ↔ (𝑌 ≼ ω → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))))
3231rspcva 3463 . . 3 ((𝑌 ∈ 𝒫 dom 𝑂 ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))) → (𝑌 ≼ ω → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌))))
3320, 24, 32syl2anc 574 . 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 197  wa 383   = wceq 1634  wcel 2148  wral 3064  Vcvv 3355  wss 3729  c0 4073  𝒫 cpw 4307   cuni 4585   class class class wbr 4797  dom cdm 5263  cres 5265  wf 6038  cfv 6042  (class class class)co 6812  ωcom 7233  cdom 8128  0cc0 10159  +∞cpnf 10294  cle 10298  [,]cicc 12402  Σ^csumge0 41102  OutMeascome 41229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-fv 6050  df-ome 41230
This theorem is referenced by:  omeunle  41256  omeiunle  41257
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