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Theorem ovolfcl 25389

Description: Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)

**Assertion**
Ref	Expression
ovolfcl	⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1^st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑁)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁))))

**Proof of Theorem ovolfcl**
Step	Hyp	Ref	Expression
1		ffvelcdm 7009	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
2	1	elin2d 4150	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ (ℝ × ℝ))
3		1st2nd2 7955	. . . 4 ⊢ ((𝐹‘𝑁) ∈ (ℝ × ℝ) → (𝐹‘𝑁) = ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
4	2, 3	syl 17	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) = ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
5	4, 1	eqeltrrd 2832	. 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
6		ancom 460	. . 3 ⊢ (((1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁)) ∧ ((1^st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑁)) ∈ ℝ)) ↔ (((1^st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑁)) ∈ ℝ) ∧ (1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁))))
7		elin 3913	. . . 4 ⊢ (⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ (⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩ ∈ ≤ ∧ ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩ ∈ (ℝ × ℝ)))
8		df-br 5087	. . . . . 6 ⊢ ((1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁)) ↔ ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩ ∈ ≤ )
9	8	bicomi 224	. . . . 5 ⊢ (⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩ ∈ ≤ ↔ (1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁)))
10		opelxp 5647	. . . . 5 ⊢ (⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩ ∈ (ℝ × ℝ) ↔ ((1^st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑁)) ∈ ℝ))
11	9, 10	anbi12i 628	. . . 4 ⊢ ((⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩ ∈ ≤ ∧ ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩ ∈ (ℝ × ℝ)) ↔ ((1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁)) ∧ ((1^st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑁)) ∈ ℝ)))
12	7, 11	bitri 275	. . 3 ⊢ (⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ ((1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁)) ∧ ((1^st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑁)) ∈ ℝ)))
13		df-3an 1088	. . 3 ⊢ (((1^st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑁)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁))) ↔ (((1^st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑁)) ∈ ℝ) ∧ (1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁))))
14	6, 12, 13	3bitr4i 303	. 2 ⊢ (⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ ((1^st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑁)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁))))
15	5, 14	sylib 218	1 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1^st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑁)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∩ cin 3896 ⟨cop 4577 class class class wbr 5086 × cxp 5609 ⟶wf 6472 ‘cfv 6476 1^st c1st 7914 2^nd c2nd 7915 ℝcr 11000 ≤ cle 11142 ℕcn 12120

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5229 ax-nul 5239 ax-pr 5365 ax-un 7663

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4279 df-if 4471 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5506 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-fv 6484 df-1st 7916 df-2nd 7917

This theorem is referenced by: ovolfioo 25390 ovolficc 25391 ovolfsval 25393 ovolfsf 25394 ovollb2lem 25411 ovolshftlem1 25432 ovolscalem1 25436 ioombl1lem1 25481 ioombl1lem3 25483 ioombl1lem4 25484 ovolfs2 25494 uniiccdif 25501 uniioovol 25502 uniioombllem2a 25505 uniioombllem2 25506 uniioombllem3a 25507 uniioombllem3 25508 uniioombllem4 25509 uniioombllem6 25511 ovolval3 46685

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