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Theorem ovolfioo 23282
Description: Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.)
Assertion
Ref Expression
ovolfioo ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
Distinct variable groups:   𝑧,𝑛,𝐴   𝑛,𝐹,𝑧

Proof of Theorem ovolfioo
StepHypRef Expression
1 ioof 12309 . . . . . 6 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
2 inss2 3867 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3 rexpssxrxp 10122 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
42, 3sstri 3645 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
5 fss 6094 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
64, 5mpan2 707 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
7 fco 6096 . . . . . 6 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
81, 6, 7sylancr 696 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
9 ffn 6083 . . . . 5 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
10 fniunfv 6545 . . . . 5 (((,) ∘ 𝐹) Fn ℕ → 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ran ((,) ∘ 𝐹))
118, 9, 103syl 18 . . . 4 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ran ((,) ∘ 𝐹))
1211sseq2d 3666 . . 3 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (𝐴 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ 𝐴 ran ((,) ∘ 𝐹)))
1312adantl 481 . 2 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ 𝐴 ran ((,) ∘ 𝐹)))
14 dfss3 3625 . . 3 (𝐴 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∀𝑧𝐴 𝑧 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛))
15 ssel2 3631 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑧𝐴) → 𝑧 ∈ ℝ)
16 eliun 4556 . . . . . . 7 (𝑧 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ 𝑧 ∈ (((,) ∘ 𝐹)‘𝑛))
17 fvco3 6314 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
18 ffvelrn 6397 . . . . . . . . . . . . . . . . 17 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
192, 18sseldi 3634 . . . . . . . . . . . . . . . 16 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (ℝ × ℝ))
20 1st2nd2 7249 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) ∈ (ℝ × ℝ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
2119, 20syl 17 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
2221fveq2d 6233 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹𝑛)) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
23 df-ov 6693 . . . . . . . . . . . . . 14 ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
2422, 23syl6eqr 2703 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹𝑛)) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
2517, 24eqtrd 2685 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
2625eleq2d 2716 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (((,) ∘ 𝐹)‘𝑛) ↔ 𝑧 ∈ ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛)))))
27 ovolfcl 23281 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
28 rexr 10123 . . . . . . . . . . . . . . 15 ((1st ‘(𝐹𝑛)) ∈ ℝ → (1st ‘(𝐹𝑛)) ∈ ℝ*)
29 rexr 10123 . . . . . . . . . . . . . . 15 ((2nd ‘(𝐹𝑛)) ∈ ℝ → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
30 elioo1 12253 . . . . . . . . . . . . . . 15 (((1st ‘(𝐹𝑛)) ∈ ℝ* ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ*) → (𝑧 ∈ ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ* ∧ (1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
3128, 29, 30syl2an 493 . . . . . . . . . . . . . 14 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ) → (𝑧 ∈ ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ* ∧ (1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
32 3anass 1059 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℝ* ∧ (1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ* ∧ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
3331, 32syl6bb 276 . . . . . . . . . . . . 13 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ) → (𝑧 ∈ ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ* ∧ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛))))))
34333adant3 1101 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → (𝑧 ∈ ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ* ∧ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛))))))
3527, 34syl 17 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ* ∧ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛))))))
3626, 35bitrd 268 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (((,) ∘ 𝐹)‘𝑛) ↔ (𝑧 ∈ ℝ* ∧ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛))))))
3736adantll 750 . . . . . . . . 9 (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (((,) ∘ 𝐹)‘𝑛) ↔ (𝑧 ∈ ℝ* ∧ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛))))))
38 rexr 10123 . . . . . . . . . . 11 (𝑧 ∈ ℝ → 𝑧 ∈ ℝ*)
3938ad2antrr 762 . . . . . . . . . 10 (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℝ*)
4039biantrurd 528 . . . . . . . . 9 (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ* ∧ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛))))))
4137, 40bitr4d 271 . . . . . . . 8 (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (((,) ∘ 𝐹)‘𝑛) ↔ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
4241rexbidva 3078 . . . . . . 7 ((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∃𝑛 ∈ ℕ 𝑧 ∈ (((,) ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
4316, 42syl5bb 272 . . . . . 6 ((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝑧 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
4415, 43sylan 487 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝑧𝐴) ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝑧 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
4544an32s 863 . . . 4 (((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑧𝐴) → (𝑧 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
4645ralbidva 3014 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∀𝑧𝐴 𝑧 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
4714, 46syl5bb 272 . 2 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
4813, 47bitr3d 270 1 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cin 3606  wss 3607  𝒫 cpw 4191  cop 4216   cuni 4468   ciun 4552   class class class wbr 4685   × cxp 5141  ran crn 5144  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  cr 9973  *cxr 10111   < clt 10112  cle 10113  cn 11058  (,)cioo 12213
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  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  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-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  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-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-ioo 12217
This theorem is referenced by:  ovollb2lem  23302  ovolunlem1  23311  ovoliunlem2  23317  ovolshftlem1  23323  ovolscalem1  23327  ioombl1lem4  23375
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