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Theorem ovolficcss 23045
 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 5559 . . 3 ran ([,] ∘ 𝐹) = ([,] “ ran 𝐹)
2 inss2 3796 . . . . . . . . . . . 12 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3 ffvelrn 6265 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹𝑦) ∈ ( ≤ ∩ (ℝ × ℝ)))
42, 3sseldi 3566 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹𝑦) ∈ (ℝ × ℝ))
5 1st2nd2 7096 . . . . . . . . . . 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 6107 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) = ([,]‘⟨(1st ‘(𝐹𝑦)), (2nd ‘(𝐹𝑦))⟩))
8 df-ov 6552 . . . . . . . . 9 ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))) = ([,]‘⟨(1st ‘(𝐹𝑦)), (2nd ‘(𝐹𝑦))⟩)
97, 8syl6eqr 2662 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) = ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))))
10 xp1st 7089 . . . . . . . . . 10 ((𝐹𝑦) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑦)) ∈ ℝ)
114, 10syl 17 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (1st ‘(𝐹𝑦)) ∈ ℝ)
12 xp2nd 7090 . . . . . . . . . 10 ((𝐹𝑦) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑦)) ∈ ℝ)
134, 12syl 17 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (2nd ‘(𝐹𝑦)) ∈ ℝ)
14 iccssre 12126 . . . . . . . . 9 (((1st ‘(𝐹𝑦)) ∈ ℝ ∧ (2nd ‘(𝐹𝑦)) ∈ ℝ) → ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))) ⊆ ℝ)
1511, 13, 14syl2anc 691 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))) ⊆ ℝ)
169, 15eqsstrd 3602 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) ⊆ ℝ)
17 reex 9906 . . . . . . . 8 ℝ ∈ V
1817elpw2 4755 . . . . . . 7 (([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹𝑦)) ⊆ ℝ)
1916, 18sylibr 223 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ)
2019ralrimiva 2949 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑦 ∈ ℕ ([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ)
21 ffn 5958 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn ℕ)
22 fveq2 6103 . . . . . . . 8 (𝑥 = (𝐹𝑦) → ([,]‘𝑥) = ([,]‘(𝐹𝑦)))
2322eleq1d 2672 . . . . . . 7 (𝑥 = (𝐹𝑦) → (([,]‘𝑥) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ))
2423ralrn 6270 . . . . . 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 246 . . . 4 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ)
27 iccf 12143 . . . . . 6 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
28 ffun 5961 . . . . . 6 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
2927, 28ax-mp 5 . . . . 5 Fun [,]
30 frn 5966 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
31 rexpssxrxp 9963 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
322, 31sstri 3577 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
3327fdmi 5965 . . . . . . 7 dom [,] = (ℝ* × ℝ*)
3432, 33sseqtr4i 3601 . . . . . 6 ( ≤ ∩ (ℝ × ℝ)) ⊆ dom [,]
3530, 34syl6ss 3580 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ dom [,])
36 funimass4 6157 . . . . 5 ((Fun [,] ∧ ran 𝐹 ⊆ dom [,]) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
3729, 35, 36sylancr 694 . . . 4 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
3826, 37mpbird 246 . . 3 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ([,] “ ran 𝐹) ⊆ 𝒫 ℝ)
391, 38syl5eqss 3612 . 2 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ)
40 sspwuni 4547 . 2 (ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ran ([,] ∘ 𝐹) ⊆ ℝ)
4139, 40sylib 207 1 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ⟨cop 4131  ∪ cuni 4372   × cxp 5036  dom cdm 5038  ran crn 5039   “ cima 5041   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  ℝcr 9814  ℝ*cxr 9952   ≤ cle 9954  ℕcn 10897  [,]cicc 12049 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-icc 12053 This theorem is referenced by:  ovollb2lem  23063  ovollb2  23064  uniiccdif  23152  uniiccvol  23154  uniioombllem3  23159  uniioombllem4  23160  uniioombllem5  23161  uniiccmbl  23164
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