Theorem ovolficcss 24073

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 6109	. . 3 ⊢ ran ([,] ∘ 𝐹) = ([,] “ ran 𝐹)
2		ffvelrn 6852	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) ∈ ( ≤ ∩ (ℝ × ℝ)))
3	2	elin2d 4179	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) ∈ (ℝ × ℝ))
4		1st2nd2 7731	. . . . . . . . . . 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 6677	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) = ([,]‘⟨(1st ‘(𝐹‘𝑦)), (2nd ‘(𝐹‘𝑦))⟩))
7		df-ov 7162	. . . . . . . . 9 ⊢ ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))) = ([,]‘⟨(1st ‘(𝐹‘𝑦)), (2nd ‘(𝐹‘𝑦))⟩)
8	6, 7	syl6eqr 2877	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) = ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))))
9		xp1st 7724	. . . . . . . . . 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 7725	. . . . . . . . . 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 12821	. . . . . . . . 9 ⊢ (((1st ‘(𝐹‘𝑦)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑦)) ∈ ℝ) → ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))) ⊆ ℝ)
14	10, 12, 13	syl2anc 586	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))) ⊆ ℝ)
15	8, 14	eqsstrd 4008	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) ⊆ ℝ)
16		reex 10631	. . . . . . . 8 ⊢ ℝ ∈ V
17	16	elpw2 5251	. . . . . . 7 ⊢ (([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹‘𝑦)) ⊆ ℝ)
18	15, 17	sylibr 236	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ)
19	18	ralrimiva 3185	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑦 ∈ ℕ ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ)
20		ffn 6517	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn ℕ)
21		fveq2 6673	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑦) → ([,]‘𝑥) = ([,]‘(𝐹‘𝑦)))
22	21	eleq1d 2900	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑦) → (([,]‘𝑥) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ))
23	22	ralrn 6857	. . . . . 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 259	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ)
26		iccf 12839	. . . . . 6 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
27		ffun 6520	. . . . . 6 ⊢ ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
28	26, 27	ax-mp 5	. . . . 5 ⊢ Fun [,]
29		frn 6523	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
30		inss2 4209	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
31		rexpssxrxp 10689	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
32	30, 31	sstri 3979	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
33	26	fdmi 6527	. . . . . . 7 ⊢ dom [,] = (ℝ* × ℝ*)
34	32, 33	sseqtrri 4007	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ dom [,]
35	29, 34	sstrdi 3982	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ dom [,])
36		funimass4 6733	. . . . 5 ⊢ ((Fun [,] ∧ ran 𝐹 ⊆ dom [,]) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
37	28, 35, 36	sylancr 589	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
38	25, 37	mpbird 259	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ([,] “ ran 𝐹) ⊆ 𝒫 ℝ)
39	1, 38	eqsstrid 4018	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ)
40		sspwuni 5025	. 2 ⊢ (ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
41	39, 40	sylib 220	1 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3141   ∩ cin 3938   ⊆ wss 3939  𝒫 cpw 4542  ⟨cop 4576  ∪ cuni 4841   × cxp 5556  dom cdm 5558  ran crn 5559   “ cima 5561   ∘ ccom 5562  Fun wfun 6352   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159  1^st c1st 7690  2^nd c2nd 7691  ℝcr 10539  ℝ^*cxr 10677   ≤ cle 10679  ℕcn 11641  [,]cicc 12744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-pre-lttri 10614  ax-pre-lttrn 10615

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-po 5477  df-so 5478  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-er 8292  df-en 8513  df-dom 8514  df-sdom 8515  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-icc 12748

This theorem is referenced by:  ovollb2lem  24092  ovollb2  24093  uniiccdif  24182  uniiccvol  24184  uniioombllem3  24189  uniioombllem4  24190  uniioombllem5  24191  uniiccmbl  24194