Theorem ovolficcss 25517

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 6274	. . 3 ⊢ ran ([,] ∘ 𝐹) = ([,] “ ran 𝐹)
2		ffvelcdm 7100	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) ∈ ( ≤ ∩ (ℝ × ℝ)))
3	2	elin2d 4214	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) ∈ (ℝ × ℝ))
4		1st2nd2 8051	. . . . . . . . . . 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 6910	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) = ([,]‘⟨(1st ‘(𝐹‘𝑦)), (2nd ‘(𝐹‘𝑦))⟩))
7		df-ov 7433	. . . . . . . . 9 ⊢ ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))) = ([,]‘⟨(1st ‘(𝐹‘𝑦)), (2nd ‘(𝐹‘𝑦))⟩)
8	6, 7	eqtr4di 2792	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) = ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))))
9		xp1st 8044	. . . . . . . . . 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 8045	. . . . . . . . . 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 13465	. . . . . . . . 9 ⊢ (((1st ‘(𝐹‘𝑦)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑦)) ∈ ℝ) → ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))) ⊆ ℝ)
14	10, 12, 13	syl2anc 584	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ((1st ‘(𝐹‘𝑦))[,](2nd ‘(𝐹‘𝑦))) ⊆ ℝ)
15	8, 14	eqsstrd 4033	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) ⊆ ℝ)
16		reex 11243	. . . . . . . 8 ⊢ ℝ ∈ V
17	16	elpw2 5339	. . . . . . 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 3143	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑦 ∈ ℕ ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ)
20		ffn 6736	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn ℕ)
21		fveq2 6906	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑦) → ([,]‘𝑥) = ([,]‘(𝐹‘𝑦)))
22	21	eleq1d 2823	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑦) → (([,]‘𝑥) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ))
23	22	ralrn 7107	. . . . . 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 13484	. . . . . 6 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
27		ffun 6739	. . . . . 6 ⊢ ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
28	26, 27	ax-mp 5	. . . . 5 ⊢ Fun [,]
29		frn 6743	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
30		inss2 4245	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
31		rexpssxrxp 11303	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
32	30, 31	sstri 4004	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
33	26	fdmi 6747	. . . . . . 7 ⊢ dom [,] = (ℝ* × ℝ*)
34	32, 33	sseqtrri 4032	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ dom [,]
35	29, 34	sstrdi 4007	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ dom [,])
36		funimass4 6972	. . . . 5 ⊢ ((Fun [,] ∧ ran 𝐹 ⊆ dom [,]) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
37	28, 35, 36	sylancr 587	. . . 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 4043	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ)
40		sspwuni 5104	. 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 1536   ∈ wcel 2105  ∀wral 3058   ∩ cin 3961   ⊆ wss 3962  𝒫 cpw 4604  ⟨cop 4636  ∪ cuni 4911   × cxp 5686  dom cdm 5688  ran crn 5689   “ cima 5691   ∘ ccom 5692  Fun wfun 6556   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  1^st c1st 8010  2^nd c2nd 8011  ℝcr 11151  ℝ^*cxr 11291   ≤ cle 11293  ℕcn 12263  [,]cicc 13386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-pre-lttri 11226  ax-pre-lttrn 11227

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-icc 13390

This theorem is referenced by:  ovollb2lem  25536  ovollb2  25537  uniiccdif  25626  uniiccvol  25628  uniioombllem3  25633  uniioombllem4  25634  uniioombllem5  25635  uniiccmbl  25638