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Theorem ovolfcl 25367

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 7053	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
2	1	elin2d 4168	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ (ℝ × ℝ))
3		1st2nd2 8007	. . . 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 2829	. 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 3930	. . . 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 5108	. . . . . 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 5674	. . . . 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 1540 ∈ wcel 2109 ∩ cin 3913 ⟨cop 4595 class class class wbr 5107 × cxp 5636 ⟶wf 6507 ‘cfv 6511 1^st c1st 7966 2^nd c2nd 7967 ℝcr 11067 ≤ cle 11209 ℕcn 12186

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-fv 6519 df-1st 7968 df-2nd 7969

This theorem is referenced by: ovolfioo 25368 ovolficc 25369 ovolfsval 25371 ovolfsf 25372 ovollb2lem 25389 ovolshftlem1 25410 ovolscalem1 25414 ioombl1lem1 25459 ioombl1lem3 25461 ioombl1lem4 25462 ovolfs2 25472 uniiccdif 25479 uniioovol 25480 uniioombllem2a 25483 uniioombllem2 25484 uniioombllem3a 25485 uniioombllem3 25486 uniioombllem4 25487 uniioombllem6 25489 ovolval3 46645

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