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Theorem ovolficcss 22962
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 5545 . . 3 ran ([,] ∘ 𝐹) = ([,] “ ran 𝐹)
2 inss2 3795 . . . . . . . . . . . 12 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3 ffvelrn 6250 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹𝑦) ∈ ( ≤ ∩ (ℝ × ℝ)))
42, 3sseldi 3565 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹𝑦) ∈ (ℝ × ℝ))
5 1st2nd2 7073 . . . . . . . . . . 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 6092 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) = ([,]‘⟨(1st ‘(𝐹𝑦)), (2nd ‘(𝐹𝑦))⟩))
8 df-ov 6530 . . . . . . . . 9 ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))) = ([,]‘⟨(1st ‘(𝐹𝑦)), (2nd ‘(𝐹𝑦))⟩)
97, 8syl6eqr 2661 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) = ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))))
10 xp1st 7066 . . . . . . . . . 10 ((𝐹𝑦) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑦)) ∈ ℝ)
114, 10syl 17 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (1st ‘(𝐹𝑦)) ∈ ℝ)
12 xp2nd 7067 . . . . . . . . . 10 ((𝐹𝑦) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑦)) ∈ ℝ)
134, 12syl 17 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (2nd ‘(𝐹𝑦)) ∈ ℝ)
14 iccssre 12082 . . . . . . . . 9 (((1st ‘(𝐹𝑦)) ∈ ℝ ∧ (2nd ‘(𝐹𝑦)) ∈ ℝ) → ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))) ⊆ ℝ)
1511, 13, 14syl2anc 690 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))) ⊆ ℝ)
169, 15eqsstrd 3601 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) ⊆ ℝ)
17 reex 9883 . . . . . . . 8 ℝ ∈ V
1817elpw2 4750 . . . . . . 7 (([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹𝑦)) ⊆ ℝ)
1916, 18sylibr 222 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ)
2019ralrimiva 2948 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑦 ∈ ℕ ([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ)
21 ffn 5944 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn ℕ)
22 fveq2 6088 . . . . . . . 8 (𝑥 = (𝐹𝑦) → ([,]‘𝑥) = ([,]‘(𝐹𝑦)))
2322eleq1d 2671 . . . . . . 7 (𝑥 = (𝐹𝑦) → (([,]‘𝑥) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ))
2423ralrn 6255 . . . . . 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 245 . . . 4 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ)
27 iccf 12099 . . . . . 6 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
28 ffun 5947 . . . . . 6 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
2927, 28ax-mp 5 . . . . 5 Fun [,]
30 frn 5952 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
31 rexpssxrxp 9940 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
322, 31sstri 3576 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
3327fdmi 5951 . . . . . . 7 dom [,] = (ℝ* × ℝ*)
3432, 33sseqtr4i 3600 . . . . . 6 ( ≤ ∩ (ℝ × ℝ)) ⊆ dom [,]
3530, 34syl6ss 3579 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ dom [,])
36 funimass4 6142 . . . . 5 ((Fun [,] ∧ ran 𝐹 ⊆ dom [,]) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
3729, 35, 36sylancr 693 . . . 4 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
3826, 37mpbird 245 . . 3 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ([,] “ ran 𝐹) ⊆ 𝒫 ℝ)
391, 38syl5eqss 3611 . 2 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ)
40 sspwuni 4541 . 2 (ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ran ([,] ∘ 𝐹) ⊆ ℝ)
4139, 40sylib 206 1 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  cin 3538  wss 3539  𝒫 cpw 4107  cop 4130   cuni 4366   × cxp 5026  dom cdm 5028  ran crn 5029  cima 5031  ccom 5032  Fun wfun 5784   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  1st c1st 7034  2nd c2nd 7035  cr 9791  *cxr 9929  cle 9931  cn 10867  [,]cicc 12005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-pre-lttri 9866  ax-pre-lttrn 9867
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-po 4949  df-so 4950  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-icc 12009
This theorem is referenced by:  ovollb2lem  22980  ovollb2  22981  uniiccdif  23069  uniiccvol  23071  uniioombllem3  23076  uniioombllem4  23077  uniioombllem5  23078  uniiccmbl  23081
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