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Theorem ovolfcl 25433

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 7033	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
2	1	elin2d 4145	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ (ℝ × ℝ))
3		1st2nd2 7981	. . . 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 3905	. . . 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 5086	. . . . . 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 5667	. . . . 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 629	. . . 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 1089	. . 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 1087 = wceq 1542 ∈ wcel 2114 ∩ cin 3888 ⟨cop 4573 class class class wbr 5085 × cxp 5629 ⟶wf 6494 ‘cfv 6498 1^st c1st 7940 2^nd c2nd 7941 ℝcr 11037 ≤ cle 11180 ℕcn 12174

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 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 3062 df-rab 3390 df-v 3431 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-fv 6506 df-1st 7942 df-2nd 7943

This theorem is referenced by: ovolfioo 25434 ovolficc 25435 ovolfsval 25437 ovolfsf 25438 ovollb2lem 25455 ovolshftlem1 25476 ovolscalem1 25480 ioombl1lem1 25525 ioombl1lem3 25527 ioombl1lem4 25528 ovolfs2 25538 uniiccdif 25545 uniioovol 25546 uniioombllem2a 25549 uniioombllem2 25550 uniioombllem3a 25551 uniioombllem3 25552 uniioombllem4 25553 uniioombllem6 25555 ovolval3 47075

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