Theorem ovolficcss 25430

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.

Step	Hyp	Ref	Expression
1		rnco2 6213	. . 3 ⊢ ran ([,] ∘ 𝐹) = ([,] “ ran 𝐹)
2		ffvelcdm 7028	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) ∈ ( ≤ ∩ (ℝ × ℝ)))
3	2	elin2d 4158	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) ∈ (ℝ × ℝ))
4		1st2nd2 7974	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑦) ∈ (ℝ × ℝ) → (𝐹‘𝑦) = ⟨(1st ‘(𝐹‘𝑦)), (2nd ‘(𝐹‘𝑦))⟩)
5	3, 4	syl 17	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) = ⟨(1st ‘(𝐹‘𝑦)), (2nd ‘(𝐹‘𝑦))⟩)
6	5	fveq2d 6839	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) = ([,]‘⟨(1st ‘(𝐹‘𝑦)), (2nd ‘(𝐹‘𝑦))⟩))
7		df-ov 7363	. . . . . . . . 9 ⊢ ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))) = ([,]‘⟨(1st ‘(𝐹‘𝑦)), (2nd ‘(𝐹‘𝑦))⟩)
8	6, 7	eqtr4di 2790	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) = ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))))
9		xp1st 7967	. . . . . . . . . 10 ⊢ ((𝐹‘𝑦) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘𝑦)) ∈ ℝ)
10	3, 9	syl 17	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (1st ‘(𝐹‘𝑦)) ∈ ℝ)
11		xp2nd 7968	. . . . . . . . . 10 ⊢ ((𝐹‘𝑦) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘𝑦)) ∈ ℝ)
12	3, 11	syl 17	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (2nd ‘(𝐹‘𝑦)) ∈ ℝ)
13		iccssre 13349	. . . . . . . . 9 ⊢ (((1st ‘(𝐹‘𝑦)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑦)) ∈ ℝ) → ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))) ⊆ ℝ)
14	10, 12, 13	syl2anc 585	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))) ⊆ ℝ)
15	8, 14	eqsstrd 3969	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) ⊆ ℝ)
16		reex 11121	. . . . . . . 8 ⊢ ℝ ∈ V
17	16	elpw2 5280	. . . . . . 7 ⊢ (([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹‘𝑦)) ⊆ ℝ)
18	15, 17	sylibr 234	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ)
19	18	ralrimiva 3129	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑦 ∈ ℕ ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ)
20		ffn 6663	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn ℕ)
21		fveq2 6835	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑦) → ([,]‘𝑥) = ([,]‘(𝐹‘𝑦)))
22	21	eleq1d 2822	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑦) → (([,]‘𝑥) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ))
23	22	ralrn 7035	. . . . . 6 ⊢ (𝐹 Fn ℕ → (∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ ↔ ∀𝑦 ∈ ℕ ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ))
24	20, 23	syl 17	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ ↔ ∀𝑦 ∈ ℕ ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ))
25	19, 24	mpbird 257	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ)
26		iccf 13368	. . . . . 6 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
27		ffun 6666	. . . . . 6 ⊢ ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
28	26, 27	ax-mp 5	. . . . 5 ⊢ Fun [,]
29		frn 6670	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
30		inss2 4191	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
31		rexpssxrxp 11181	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
32	30, 31	sstri 3944	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
33	26	fdmi 6674	. . . . . . 7 ⊢ dom [,] = (ℝ* × ℝ*)
34	32, 33	sseqtrri 3984	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ dom [,]
35	29, 34	sstrdi 3947	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ dom [,])
36		funimass4 6899	. . . . 5 ⊢ ((Fun [,] ∧ ran 𝐹 ⊆ dom [,]) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
37	28, 35, 36	sylancr 588	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
38	25, 37	mpbird 257	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ([,] “ ran 𝐹) ⊆ 𝒫 ℝ)
39	1, 38	eqsstrid 3973	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ)
40		sspwuni 5056	. 2 ⊢ (ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
41	39, 40	sylib 218	1 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4555  ⟨cop 4587  ∪ cuni 4864   × cxp 5623  dom cdm 5625  ran crn 5626   “ cima 5628   ∘ ccom 5629  Fun wfun 6487   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  ℝcr 11029  ℝ^*cxr 11169   ≤ cle 11171  ℕcn 12149  [,]cicc 13268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-pre-lttri 11104  ax-pre-lttrn 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-po 5533  df-so 5534  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-icc 13272

This theorem is referenced by:  ovollb2lem  25449  ovollb2  25450  uniiccdif  25539  uniiccvol  25541  uniioombllem3  25546  uniioombllem4  25547  uniioombllem5  25548  uniiccmbl  25551