Theorem ovolficcss 23238

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 5642	. . 3 ⊢ ran ([,] ∘ 𝐹) = ([,] “ ran 𝐹)
2		inss2 3834	. . . . . . . . . . . 12 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3		ffvelrn 6357	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) ∈ ( ≤ ∩ (ℝ × ℝ)))
4	2, 3	sseldi 3601	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) ∈ (ℝ × ℝ))
5		1st2nd2 7205	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑦) ∈ (ℝ × ℝ) → (𝐹‘𝑦) = ⟨(1st ‘(𝐹‘𝑦)), (2nd ‘(𝐹‘𝑦))⟩)
6	4, 5	syl 17	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) = ⟨(1st ‘(𝐹‘𝑦)), (2nd ‘(𝐹‘𝑦))⟩)
7	6	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) = ([,]‘⟨(1st ‘(𝐹‘𝑦)), (2nd ‘(𝐹‘𝑦))⟩))
8		df-ov 6653	. . . . . . . . 9 ⊢ ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))) = ([,]‘⟨(1st ‘(𝐹‘𝑦)), (2nd ‘(𝐹‘𝑦))⟩)
9	7, 8	syl6eqr 2674	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) = ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))))
10		xp1st 7198	. . . . . . . . . 10 ⊢ ((𝐹‘𝑦) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘𝑦)) ∈ ℝ)
11	4, 10	syl 17	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (1st ‘(𝐹‘𝑦)) ∈ ℝ)
12		xp2nd 7199	. . . . . . . . . 10 ⊢ ((𝐹‘𝑦) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘𝑦)) ∈ ℝ)
13	4, 12	syl 17	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (2nd ‘(𝐹‘𝑦)) ∈ ℝ)
14		iccssre 12255	. . . . . . . . 9 ⊢ (((1st ‘(𝐹‘𝑦)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑦)) ∈ ℝ) → ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))) ⊆ ℝ)
15	11, 13, 14	syl2anc 693	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))) ⊆ ℝ)
16	9, 15	eqsstrd 3639	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) ⊆ ℝ)
17		reex 10027	. . . . . . . 8 ⊢ ℝ ∈ V
18	17	elpw2 4828	. . . . . . 7 ⊢ (([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹‘𝑦)) ⊆ ℝ)
19	16, 18	sylibr 224	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ)
20	19	ralrimiva 2966	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑦 ∈ ℕ ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ)
21		ffn 6045	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn ℕ)
22		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑦) → ([,]‘𝑥) = ([,]‘(𝐹‘𝑦)))
23	22	eleq1d 2686	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑦) → (([,]‘𝑥) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ))
24	23	ralrn 6362	. . . . . 6 ⊢ (𝐹 Fn ℕ → (∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ ↔ ∀𝑦 ∈ ℕ ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ))
25	21, 24	syl 17	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ ↔ ∀𝑦 ∈ ℕ ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ))
26	20, 25	mpbird 247	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ)
27		iccf 12272	. . . . . 6 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
28		ffun 6048	. . . . . 6 ⊢ ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
29	27, 28	ax-mp 5	. . . . 5 ⊢ Fun [,]
30		frn 6053	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
31		rexpssxrxp 10084	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
32	2, 31	sstri 3612	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
33	27	fdmi 6052	. . . . . . 7 ⊢ dom [,] = (ℝ* × ℝ*)
34	32, 33	sseqtr4i 3638	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ dom [,]
35	30, 34	syl6ss 3615	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ dom [,])
36		funimass4 6247	. . . . 5 ⊢ ((Fun [,] ∧ ran 𝐹 ⊆ dom [,]) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
37	29, 35, 36	sylancr 695	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
38	26, 37	mpbird 247	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ([,] “ ran 𝐹) ⊆ 𝒫 ℝ)
39	1, 38	syl5eqss 3649	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ)
40		sspwuni 4611	. 2 ⊢ (ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
41	39, 40	sylib 208	1 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  ⟨cop 4183  ∪ cuni 4436   × cxp 5112  dom cdm 5114  ran crn 5115   “ cima 5117   ∘ ccom 5118  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  ℝcr 9935  ℝ^*cxr 10073   ≤ cle 10075  ℕcn 11020  [,]cicc 12178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-icc 12182

This theorem is referenced by:  ovollb2lem  23256  ovollb2  23257  uniiccdif  23346  uniiccvol  23348  uniioombllem3  23353  uniioombllem4  23354  uniioombllem5  23355  uniiccmbl  23358