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Theorem ovolficc 25516
Description: Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)
Assertion
Ref Expression
ovolficc ((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 ovolficc
StepHypRef Expression
1 iccf 13484 . . . . . 6 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
2 inss2 4245 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3 rexpssxrxp 11303 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
42, 3sstri 4004 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
5 fss 6752 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
64, 5mpan2 691 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
7 fco 6760 . . . . . 6 (([,]:(ℝ* × ℝ*)⟶𝒫 ℝ*𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹):ℕ⟶𝒫 ℝ*)
81, 6, 7sylancr 587 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ([,] ∘ 𝐹):ℕ⟶𝒫 ℝ*)
9 ffn 6736 . . . . 5 (([,] ∘ 𝐹):ℕ⟶𝒫 ℝ* → ([,] ∘ 𝐹) Fn ℕ)
10 fniunfv 7266 . . . . 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 4027 . . 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 3983 . . 3 (𝐴 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∀𝑧𝐴 𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛))
15 ssel2 3989 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑧𝐴) → 𝑧 ∈ ℝ)
16 eliun 4999 . . . . . . 7 (𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ 𝑧 ∈ (([,] ∘ 𝐹)‘𝑛))
17 simpll 767 . . . . . . . . 9 (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℝ)
18 fvco3 7007 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑛) = ([,]‘(𝐹𝑛)))
19 ffvelcdm 7100 . . . . . . . . . . . . . . . . 17 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
2019elin2d 4214 . . . . . . . . . . . . . . . 16 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (ℝ × ℝ))
21 1st2nd2 8051 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) ∈ (ℝ × ℝ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
2220, 21syl 17 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
2322fveq2d 6910 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ([,]‘(𝐹𝑛)) = ([,]‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
24 df-ov 7433 . . . . . . . . . . . . . 14 ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))) = ([,]‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
2523, 24eqtr4di 2792 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ([,]‘(𝐹𝑛)) = ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))))
2618, 25eqtrd 2774 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑛) = ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))))
2726eleq2d 2824 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ 𝑧 ∈ ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛)))))
28 ovolfcl 25514 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
29 elicc2 13448 . . . . . . . . . . . . . 14 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ) → (𝑧 ∈ ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
30 3anass 1094 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ ∧ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
3129, 30bitrdi 287 . . . . . . . . . . . . 13 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ) → (𝑧 ∈ ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ ∧ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛))))))
32313adant3 1131 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → (𝑧 ∈ ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ ∧ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛))))))
3328, 32syl 17 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ ∧ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛))))))
3427, 33bitrd 279 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ (𝑧 ∈ ℝ ∧ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛))))))
3534adantll 714 . . . . . . . . 9 (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ (𝑧 ∈ ℝ ∧ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛))))))
3617, 35mpbirand 707 . . . . . . . 8 (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
3736rexbidva 3174 . . . . . . 7 ((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∃𝑛 ∈ ℕ 𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
3816, 37bitrid 283 . . . . . 6 ((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
3915, 38sylan 580 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝑧𝐴) ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
4039an32s 652 . . . 4 (((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑧𝐴) → (𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
4140ralbidva 3173 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∀𝑧𝐴 𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
4214, 41bitrid 283 . 2 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
4313, 42bitr3d 281 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 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  cin 3961  wss 3962  𝒫 cpw 4604  cop 4636   cuni 4911   ciun 4995   class class class wbr 5147   × cxp 5686  ran crn 5689  ccom 5692   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  cr 11151  *cxr 11291  cle 11293  cn 12263  [,]cicc 13386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-pre-lttri 11226  ax-pre-lttrn 11227
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-icc 13390
This theorem is referenced by:  ovollb2lem  25536  ovolctb  25538  ovolicc1  25564  ioombl1lem4  25609
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