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Theorem ovolfcl 25780

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 7079	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
2	1	elin2d 4151	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ (ℝ × ℝ))
3		1st2nd2 8038	. . . 4 ⊢ ((𝐹‘𝑁) ∈ (ℝ × ℝ) → (𝐹‘𝑁) = ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
4	2, 3	syl 18	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) = ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
5	4, 1	eqeltrrd 2862	. 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
6		ancom 466	. . 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 3915	. . . 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 5104	. . . . . 6 ⊢ ((1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁)) ↔ ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩ ∈ ≤ )
9	8	bicomi 227	. . . . 5 ⊢ (⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩ ∈ ≤ ↔ (1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁)))
10		opelxp 5687	. . . . 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 640	. . . 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 278	. . 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 1105	. . 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 306	. 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 221	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∩ cin 3898 ⟨cop 4590 class class class wbr 5103 × cxp 5649 ⟶wf 6533 ‘cfv 6537 1^st c1st 7997 2^nd c2nd 7998 ℝcr 11192 ≤ cle 11337 ℕcn 12328

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-pr 5391 ax-un 7749

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 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-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-1st 7999 df-2nd 8000

This theorem is used by: ovolfioo 25781 ovolficc 25782 ovolfsval 25784 ovolfsf 25785 ovollb2lem 25802 ovolshftlem1 25823 ovolscalem1 25827 ioombl1lem1 25872 ioombl1lem3 25874 ioombl1lem4 25875 ovolfs2 25885 uniiccdif 25892 uniioovol 25893 uniioombllem2a 25896 uniioombllem2 25897 uniioombllem3a 25898 uniioombllem3 25899 uniioombllem4 25900 uniioombllem6 25902 ovolval3 47626

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