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Theorem ovolfcl 25423

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 7026	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
2	1	elin2d 4157	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ (ℝ × ℝ))
3		1st2nd2 7972	. . . 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 2837	. 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 3917	. . . 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 5099	. . . . . 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 5660	. . . . 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 2113 ∩ cin 3900 ⟨cop 4586 class class class wbr 5098 × cxp 5622 ⟶wf 6488 ‘cfv 6492 1^st c1st 7931 2^nd c2nd 7932 ℝcr 11025 ≤ cle 11167 ℕcn 12145

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-1st 7933 df-2nd 7934

This theorem is referenced by: ovolfioo 25424 ovolficc 25425 ovolfsval 25427 ovolfsf 25428 ovollb2lem 25445 ovolshftlem1 25466 ovolscalem1 25470 ioombl1lem1 25515 ioombl1lem3 25517 ioombl1lem4 25518 ovolfs2 25528 uniiccdif 25535 uniioovol 25536 uniioombllem2a 25539 uniioombllem2 25540 uniioombllem3a 25541 uniioombllem3 25542 uniioombllem4 25543 uniioombllem6 25545 ovolval3 46891

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