ovolficcss - Metamath Proof Explorer

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Theorem ovolficcss 23673

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 5896	. . 3 ⊢ ran ([,] ∘ 𝐹) = ([,] “ ran 𝐹)
2		inss2 4053	. . . . . . . . . . . 12 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3		ffvelrn 6621	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) ∈ ( ≤ ∩ (ℝ × ℝ)))
4	2, 3	sseldi 3818	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) ∈ (ℝ × ℝ))
5		1st2nd2 7484	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑦) ∈ (ℝ × ℝ) → (𝐹‘𝑦) = ⟨(1^st ‘(𝐹‘𝑦)), (2^nd ‘(𝐹‘𝑦))⟩)
6	4, 5	syl 17	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) = ⟨(1^st ‘(𝐹‘𝑦)), (2^nd ‘(𝐹‘𝑦))⟩)
7	6	fveq2d 6450	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) = ([,]‘⟨(1^st ‘(𝐹‘𝑦)), (2^nd ‘(𝐹‘𝑦))⟩))
8		df-ov 6925	. . . . . . . . 9 ⊢ ((1^st ‘(𝐹‘𝑦))[,](2^nd ‘(𝐹‘𝑦))) = ([,]‘⟨(1^st ‘(𝐹‘𝑦)), (2^nd ‘(𝐹‘𝑦))⟩)
9	7, 8	syl6eqr 2831	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) = ((1^st ‘(𝐹‘𝑦))[,](2^nd ‘(𝐹‘𝑦))))
10		xp1st 7477	. . . . . . . . . 10 ⊢ ((𝐹‘𝑦) ∈ (ℝ × ℝ) → (1^st ‘(𝐹‘𝑦)) ∈ ℝ)
11	4, 10	syl 17	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (1^st ‘(𝐹‘𝑦)) ∈ ℝ)
12		xp2nd 7478	. . . . . . . . . 10 ⊢ ((𝐹‘𝑦) ∈ (ℝ × ℝ) → (2^nd ‘(𝐹‘𝑦)) ∈ ℝ)
13	4, 12	syl 17	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → (2^nd ‘(𝐹‘𝑦)) ∈ ℝ)
14		iccssre 12567	. . . . . . . . 9 ⊢ (((1^st ‘(𝐹‘𝑦)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑦)) ∈ ℝ) → ((1^st ‘(𝐹‘𝑦))[,](2^nd ‘(𝐹‘𝑦))) ⊆ ℝ)
15	11, 13, 14	syl2anc 579	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ((1^st ‘(𝐹‘𝑦))[,](2^nd ‘(𝐹‘𝑦))) ⊆ ℝ)
16	9, 15	eqsstrd 3857	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) ⊆ ℝ)
17		reex 10363	. . . . . . . 8 ⊢ ℝ ∈ V
18	17	elpw2 5062	. . . . . . 7 ⊢ (([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹‘𝑦)) ⊆ ℝ)
19	16, 18	sylibr 226	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑦 ∈ ℕ) → ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ)
20	19	ralrimiva 3147	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑦 ∈ ℕ ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ)
21		ffn 6291	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn ℕ)
22		fveq2 6446	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑦) → ([,]‘𝑥) = ([,]‘(𝐹‘𝑦)))
23	22	eleq1d 2843	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑦) → (([,]‘𝑥) ∈ 𝒫 ℝ ↔ ([,]‘(𝐹‘𝑦)) ∈ 𝒫 ℝ))
24	23	ralrn 6626	. . . . . 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 249	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ)
27		iccf 12585	. . . . . 6 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
28		ffun 6294	. . . . . 6 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → Fun [,])
29	27, 28	ax-mp 5	. . . . 5 ⊢ Fun [,]
30		frn 6297	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
31		rexpssxrxp 10421	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
32	2, 31	sstri 3829	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
33	27	fdmi 6301	. . . . . . 7 ⊢ dom [,] = (ℝ^* × ℝ^*)
34	32, 33	sseqtr4i 3856	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ dom [,]
35	30, 34	syl6ss 3832	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ dom [,])
36		funimass4 6507	. . . . 5 ⊢ ((Fun [,] ∧ ran 𝐹 ⊆ dom [,]) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
37	29, 35, 36	sylancr 581	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (([,] “ ran 𝐹) ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ ran 𝐹([,]‘𝑥) ∈ 𝒫 ℝ))
38	26, 37	mpbird 249	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ([,] “ ran 𝐹) ⊆ 𝒫 ℝ)
39	1, 38	syl5eqss 3867	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ)
40		sspwuni 4845	. 2 ⊢ (ran ([,] ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
41	39, 40	sylib 210	1 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∀wral 3089 ∩ cin 3790 ⊆ wss 3791 𝒫 cpw 4378 ⟨cop 4403 ∪ cuni 4671 × cxp 5353 dom cdm 5355 ran crn 5356 “ cima 5358 ∘ ccom 5359 Fun wfun 6129 Fn wfn 6130 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 1^st c1st 7443 2^nd c2nd 7444 ℝcr 10271 ℝ^*cxr 10410 ≤ cle 10412 ℕcn 11374 [,]cicc 12490

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-pre-lttri 10346 ax-pre-lttrn 10347

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-icc 12494

This theorem is referenced by: ovollb2lem 23692 ovollb2 23693 uniiccdif 23782 uniiccvol 23784 uniioombllem3 23789 uniioombllem4 23790 uniioombllem5 23791 uniiccmbl 23794

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