MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolficcss Structured version   Visualization version   GIF version

Theorem ovolficcss 23438
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 5803 . . 3 ran ([,] ∘ 𝐹) = ([,] “ ran 𝐹)
2 inss2 3977 . . . . . . . . . . . 12 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3 ffvelrn 6520 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹𝑦) ∈ ( ≤ ∩ (ℝ × ℝ)))
42, 3sseldi 3742 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹𝑦) ∈ (ℝ × ℝ))
5 1st2nd2 7372 . . . . . . . . . . 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 6356 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) = ([,]‘⟨(1st ‘(𝐹𝑦)), (2nd ‘(𝐹𝑦))⟩))
8 df-ov 6816 . . . . . . . . 9 ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))) = ([,]‘⟨(1st ‘(𝐹𝑦)), (2nd ‘(𝐹𝑦))⟩)
97, 8syl6eqr 2812 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) = ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))))
10 xp1st 7365 . . . . . . . . . 10 ((𝐹𝑦) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑦)) ∈ ℝ)
114, 10syl 17 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (1st ‘(𝐹𝑦)) ∈ ℝ)
12 xp2nd 7366 . . . . . . . . . 10 ((𝐹𝑦) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑦)) ∈ ℝ)
134, 12syl 17 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (2nd ‘(𝐹𝑦)) ∈ ℝ)
14 iccssre 12448 . . . . . . . . 9 (((1st ‘(𝐹𝑦)) ∈ ℝ ∧ (2nd ‘(𝐹𝑦)) ∈ ℝ) → ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))) ⊆ ℝ)
1511, 13, 14syl2anc 696 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ((1st ‘(𝐹𝑦))[,](2nd ‘(𝐹𝑦))) ⊆ ℝ)
169, 15eqsstrd 3780 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) ⊆ ℝ)
17 reex 10219 . . . . . . . 8 ℝ ∈ V
1817elpw2 4977 . . . . . . 7 (([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹𝑦)) ⊆ ℝ)
1916, 18sylibr 224 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ)
2019ralrimiva 3104 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑦 ∈ ℕ ([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ)
21 ffn 6206 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn ℕ)
22 fveq2 6352 . . . . . . . 8 (𝑥 = (𝐹𝑦) → ([,]‘𝑥) = ([,]‘(𝐹𝑦)))
2322eleq1d 2824 . . . . . . 7 (𝑥 = (𝐹𝑦) → (([,]‘𝑥) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹𝑦)) ∈ 𝒫 ℝ))
2423ralrn 6525 . . . . . 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 12465 . . . . . 6 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
28 ffun 6209 . . . . . 6 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
2927, 28ax-mp 5 . . . . 5 Fun [,]
30 frn 6214 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
31 rexpssxrxp 10276 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
322, 31sstri 3753 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
3327fdmi 6213 . . . . . . 7 dom [,] = (ℝ* × ℝ*)
3432, 33sseqtr4i 3779 . . . . . 6 ( ≤ ∩ (ℝ × ℝ)) ⊆ dom [,]
3530, 34syl6ss 3756 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ dom [,])
36 funimass4 6409 . . . . 5 ((Fun [,] ∧ ran 𝐹 ⊆ dom [,]) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
3729, 35, 36sylancr 698 . . . 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 3790 . 2 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ)
40 sspwuni 4763 . 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 1632  wcel 2139  wral 3050  cin 3714  wss 3715  𝒫 cpw 4302  cop 4327   cuni 4588   × cxp 5264  dom cdm 5266  ran crn 5267  cima 5269  ccom 5270  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  cr 10127  *cxr 10265  cle 10267  cn 11212  [,]cicc 12371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-pre-lttri 10202  ax-pre-lttrn 10203
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-icc 12375
This theorem is referenced by:  ovollb2lem  23456  ovollb2  23457  uniiccdif  23546  uniiccvol  23548  uniioombllem3  23553  uniioombllem4  23554  uniioombllem5  23555  uniiccmbl  23558
  Copyright terms: Public domain W3C validator