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Theorem ovolficc 24365
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 13036 . . . . . 6 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
2 inss2 4144 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3 rexpssxrxp 10878 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
42, 3sstri 3910 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
5 fss 6562 . . . . . . 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 6569 . . . . . 6 (([,]:(ℝ* × ℝ*)⟶𝒫 ℝ*𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹):ℕ⟶𝒫 ℝ*)
81, 6, 7sylancr 590 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ([,] ∘ 𝐹):ℕ⟶𝒫 ℝ*)
9 ffn 6545 . . . . 5 (([,] ∘ 𝐹):ℕ⟶𝒫 ℝ* → ([,] ∘ 𝐹) Fn ℕ)
10 fniunfv 7060 . . . . 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 3933 . . 3 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (𝐴 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ 𝐴 ran ([,] ∘ 𝐹)))
1312adantl 485 . 2 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ 𝐴 ran ([,] ∘ 𝐹)))
14 dfss3 3888 . . 3 (𝐴 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∀𝑧𝐴 𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛))
15 ssel2 3895 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑧𝐴) → 𝑧 ∈ ℝ)
16 eliun 4908 . . . . . . 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 6810 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑛) = ([,]‘(𝐹𝑛)))
19 ffvelrn 6902 . . . . . . . . . . . . . . . . 17 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
2019elin2d 4113 . . . . . . . . . . . . . . . 16 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (ℝ × ℝ))
21 1st2nd2 7800 . . . . . . . . . . . . . . . 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 6721 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ([,]‘(𝐹𝑛)) = ([,]‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
24 df-ov 7216 . . . . . . . . . . . . . 14 ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))) = ([,]‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
2523, 24eqtr4di 2796 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ([,]‘(𝐹𝑛)) = ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))))
2618, 25eqtrd 2777 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑛) = ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))))
2726eleq2d 2823 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ 𝑧 ∈ ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛)))))
28 ovolfcl 24363 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
29 elicc2 13000 . . . . . . . . . . . . . 14 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ) → (𝑧 ∈ ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
30 3anass 1097 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ ∧ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
3129, 30bitrdi 290 . . . . . . . . . . . . 13 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ) → (𝑧 ∈ ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ ∧ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛))))))
32313adant3 1134 . . . . . . . . . . . 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 282 . . . . . . . . . 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 3215 . . . . . . 7 ((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∃𝑛 ∈ ℕ 𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
3816, 37syl5bb 286 . . . . . 6 ((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
3915, 38sylan 583 . . . . 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 3117 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∀𝑧𝐴 𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
4214, 41syl5bb 286 . 2 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
4313, 42bitr3d 284 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 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wrex 3062  cin 3865  wss 3866  𝒫 cpw 4513  cop 4547   cuni 4819   ciun 4904   class class class wbr 5053   × cxp 5549  ran crn 5552  ccom 5555   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  1st c1st 7759  2nd c2nd 7760  cr 10728  *cxr 10866  cle 10868  cn 11830  [,]cicc 12938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-pre-lttri 10803  ax-pre-lttrn 10804
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-po 5468  df-so 5469  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-icc 12942
This theorem is referenced by:  ovollb2lem  24385  ovolctb  24387  ovolicc1  24413  ioombl1lem4  24458
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