ovolficc - Metamath Proof Explorer

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Theorem ovolficc 25782

Description: Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)

**Assertion**
Ref	Expression
ovolficc	⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ([,] ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛)))))

Distinct variable groups: 𝑧,𝑛,𝐴 𝑛,𝐹,𝑧

**Proof of Theorem ovolficc**
Step	Hyp	Ref	Expression
1		iccf 13572	. . . . . 6 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
2		inss2 4183	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3		rexpssxrxp 11347	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
4	2, 3	sstri 3940	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
5		fss 6724	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^)) → 𝐹:ℕ⟶(ℝ^ × ℝ^*))
6	4, 5	mpan2 704	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
7		fco 6732	. . . . . 6 ⊢ (([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ ∧ 𝐹:ℕ⟶(ℝ^* × ℝ^)) → ([,] ∘ 𝐹):ℕ⟶𝒫 ℝ^)
8	1, 6, 7	sylancr 599	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ([,] ∘ 𝐹):ℕ⟶𝒫 ℝ^*)
9		ffn 6707	. . . . 5 ⊢ (([,] ∘ 𝐹):ℕ⟶𝒫 ℝ^* → ([,] ∘ 𝐹) Fn ℕ)
10		fniunfv 7249	. . . . 5 ⊢ (([,] ∘ 𝐹) Fn ℕ → ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) = ∪ ran ([,] ∘ 𝐹))
11	8, 9, 10	3syl 19	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) = ∪ ran ([,] ∘ 𝐹))
12	11	sseq2d 3963	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)))
13	12	adantl 487	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)))
14		dfss3 3920	. . 3 ⊢ (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛))
15		ssel2 3926	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
16		eliun 4955	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ 𝑧 ∈ (([,] ∘ 𝐹)‘𝑛))
17		simpll 779	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℝ)
18		fvco3 6983	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑛) = ([,]‘(𝐹‘𝑛)))
19		ffvelcdm 7079	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
20	19	elin2d 4151	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ × ℝ))
21		1st2nd2 8038	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑛) ∈ (ℝ × ℝ) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
22	20, 21	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
23	22	fveq2d 6887	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ([,]‘(𝐹‘𝑛)) = ([,]‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
24		df-ov 7421	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘(𝐹‘𝑛))[,](2^nd ‘(𝐹‘𝑛))) = ([,]‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
25	23, 24	eqtr4di 2814	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ([,]‘(𝐹‘𝑛)) = ((1^st ‘(𝐹‘𝑛))[,](2^nd ‘(𝐹‘𝑛))))
26	18, 25	eqtrd 2796	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑛) = ((1^st ‘(𝐹‘𝑛))[,](2^nd ‘(𝐹‘𝑛))))
27	26	eleq2d 2847	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ 𝑧 ∈ ((1^st ‘(𝐹‘𝑛))[,](2^nd ‘(𝐹‘𝑛)))))
28		ovolfcl 25780	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1^st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))))
29		elicc2 13535	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑛)) ∈ ℝ) → (𝑧 ∈ ((1^st ‘(𝐹‘𝑛))[,](2^nd ‘(𝐹‘𝑛))) ↔ (𝑧 ∈ ℝ ∧ (1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛)))))
30		3anass 1111	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℝ ∧ (1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛))) ↔ (𝑧 ∈ ℝ ∧ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛)))))
31	29, 30	bitrdi 290	. . . . . . . . . . . . 13 ⊢ (((1^st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑛)) ∈ ℝ) → (𝑧 ∈ ((1^st ‘(𝐹‘𝑛))[,](2^nd ‘(𝐹‘𝑛))) ↔ (𝑧 ∈ ℝ ∧ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛))))))
32	31	3adant3 1150	. . . . . . . . . . . 12 ⊢ (((1^st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))) → (𝑧 ∈ ((1^st ‘(𝐹‘𝑛))[,](2^nd ‘(𝐹‘𝑛))) ↔ (𝑧 ∈ ℝ ∧ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛))))))
33	28, 32	syl 18	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ ((1^st ‘(𝐹‘𝑛))[,](2^nd ‘(𝐹‘𝑛))) ↔ (𝑧 ∈ ℝ ∧ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛))))))
34	27, 33	bitrd 282	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ (𝑧 ∈ ℝ ∧ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛))))))
35	34	adantll 727	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ (𝑧 ∈ ℝ ∧ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛))))))
36	17, 35	mpbirand 720	. . . . . . . 8 ⊢ (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛)))))
37	36	rexbidva 3185	. . . . . . 7 ⊢ ((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∃𝑛 ∈ ℕ 𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛)))))
38	16, 37	bitrid 286	. . . . . 6 ⊢ ((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝑧 ∈ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛)))))
39	15, 38	sylan 592	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝑧 ∈ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛)))))
40	39	an32s 665	. . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛)))))
41	40	ralbidva 3184	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛)))))
42	14, 41	bitrid 286	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛)))))
43	13, 42	bitr3d 284	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ([,] ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛)))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3077 ∃wrex 3087 ∩ cin 3898 ⊆ wss 3899 𝒫 cpw 4557 ⟨cop 4590 ∪ cuni 4867 ∪ ciun 4951 class class class wbr 5103 × cxp 5649 ran crn 5652 ∘ ccom 5655 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 1^st c1st 7997 2^nd c2nd 7998 ℝcr 11192 ℝ^*cxr 11335 ≤ cle 11337 ℕcn 12328 [,]cicc 13472

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-pre-lttri 11267 ax-pre-lttrn 11268

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7421 df-oprab 7422 df-mpo 7423 df-1st 7999 df-2nd 8000 df-er 8710 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-icc 13476

This theorem is used by: ovollb2lem 25802 ovolctb 25804 ovolicc1 25830 ioombl1lem4 25875

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