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Theorem ovolficcss 23284
Description: Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.)
Assertion
Ref Expression
ovolficcss (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)

Proof of Theorem ovolficcss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnco2 5680 . . 3 ran ([,] ∘ 𝐹) = ([,] “ ran 𝐹)
2 inss2 3867 . . . . . . . . . . . 12 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3 ffvelrn 6397 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹𝑦) ∈ ( ≤ ∩ (ℝ × ℝ)))
42, 3sseldi 3634 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹𝑦) ∈ (ℝ × ℝ))
5 1st2nd2 7249 . . . . . . . . . . 11 ((𝐹𝑦) ∈ (ℝ × ℝ) → (𝐹𝑦) = ⟨(1st ‘(𝐹𝑦)), (2nd ‘(𝐹𝑦))⟩)
64, 5syl 17 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹𝑦) = ⟨(1st ‘(𝐹𝑦)), (2nd ‘(𝐹𝑦))⟩)
76fveq2d 6233 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) = ([,]‘⟨(1st ‘(𝐹𝑦)), (2nd ‘(𝐹𝑦))⟩))
8 df-ov 6693 . . . . . . . . 9 ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))) = ([,]‘⟨(1st ‘(𝐹𝑦)), (2nd ‘(𝐹𝑦))⟩)
97, 8syl6eqr 2703 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) = ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))))
10 xp1st 7242 . . . . . . . . . 10 ((𝐹𝑦) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑦)) ∈ ℝ)
114, 10syl 17 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (1st ‘(𝐹𝑦)) ∈ ℝ)
12 xp2nd 7243 . . . . . . . . . 10 ((𝐹𝑦) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑦)) ∈ ℝ)
134, 12syl 17 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (2nd ‘(𝐹𝑦)) ∈ ℝ)
14 iccssre 12293 . . . . . . . . 9 (((1st ‘(𝐹𝑦)) ∈ ℝ ∧ (2nd ‘(𝐹𝑦)) ∈ ℝ) → ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))) ⊆ ℝ)
1511, 13, 14syl2anc 694 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))) ⊆ ℝ)
169, 15eqsstrd 3672 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) ⊆ ℝ)
17 reex 10065 . . . . . . . 8 ℝ ∈ V
1817elpw2 4858 . . . . . . 7 (([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹𝑦)) ⊆ ℝ)
1916, 18sylibr 224 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ)
2019ralrimiva 2995 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑦 ∈ ℕ ([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ)
21 ffn 6083 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn ℕ)
22 fveq2 6229 . . . . . . . 8 (𝑥 = (𝐹𝑦) → ([,]‘𝑥) = ([,]‘(𝐹𝑦)))
2322eleq1d 2715 . . . . . . 7 (𝑥 = (𝐹𝑦) → (([,]‘𝑥) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ))
2423ralrn 6402 . . . . . 6 (𝐹 Fn ℕ → (∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ ↔ ∀𝑦 ∈ ℕ ([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ))
2521, 24syl 17 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ ↔ ∀𝑦 ∈ ℕ ([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ))
2620, 25mpbird 247 . . . 4 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ)
27 iccf 12310 . . . . . 6 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
28 ffun 6086 . . . . . 6 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
2927, 28ax-mp 5 . . . . 5 Fun [,]
30 frn 6091 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
31 rexpssxrxp 10122 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
322, 31sstri 3645 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
3327fdmi 6090 . . . . . . 7 dom [,] = (ℝ* × ℝ*)
3432, 33sseqtr4i 3671 . . . . . 6 ( ≤ ∩ (ℝ × ℝ)) ⊆ dom [,]
3530, 34syl6ss 3648 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ dom [,])
36 funimass4 6286 . . . . 5 ((Fun [,] ∧ ran 𝐹 ⊆ dom [,]) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
3729, 35, 36sylancr 696 . . . 4 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
3826, 37mpbird 247 . . 3 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ([,] “ ran 𝐹) ⊆ 𝒫 ℝ)
391, 38syl5eqss 3682 . 2 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ)
40 sspwuni 4643 . 2 (ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ran ([,] ∘ 𝐹) ⊆ ℝ)
4139, 40sylib 208 1 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  cin 3606  wss 3607  𝒫 cpw 4191  cop 4216   cuni 4468   × cxp 5141  dom cdm 5143  ran crn 5144  cima 5146  ccom 5147  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  cr 9973  *cxr 10111  cle 10113  cn 11058  [,]cicc 12216
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-icc 12220
This theorem is referenced by:  ovollb2lem  23302  ovollb2  23303  uniiccdif  23392  uniiccvol  23394  uniioombllem3  23399  uniioombllem4  23400  uniioombllem5  23401  uniiccmbl  23404
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