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Theorem ovolfcl 25501
Description: Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)
Assertion
Ref Expression
ovolfcl ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))

Proof of Theorem ovolfcl
StepHypRef Expression
1 ffvelcdm 7101 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
21elin2d 4205 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ (ℝ × ℝ))
3 1st2nd2 8053 . . . 4 ((𝐹𝑁) ∈ (ℝ × ℝ) → (𝐹𝑁) = ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩)
42, 3syl 17 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) = ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩)
54, 1eqeltrrd 2842 . 2 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
6 ancom 460 . . 3 (((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ∧ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ)) ↔ (((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ) ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
7 elin 3967 . . . 4 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ ∧ ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ (ℝ × ℝ)))
8 df-br 5144 . . . . . 6 ((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ↔ ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ )
98bicomi 224 . . . . 5 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ ↔ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)))
10 opelxp 5721 . . . . 5 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ (ℝ × ℝ) ↔ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ))
119, 10anbi12i 628 . . . 4 ((⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ ∧ ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ (ℝ × ℝ)) ↔ ((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ∧ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ)))
127, 11bitri 275 . . 3 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ ((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ∧ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ)))
13 df-3an 1089 . . 3 (((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))) ↔ (((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ) ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
146, 12, 133bitr4i 303 . 2 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
155, 14sylib 218 1 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  cin 3950  cop 4632   class class class wbr 5143   × cxp 5683  wf 6557  cfv 6561  1st c1st 8012  2nd c2nd 8013  cr 11154  cle 11296  cn 12266
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-1st 8014  df-2nd 8015
This theorem is referenced by:  ovolfioo  25502  ovolficc  25503  ovolfsval  25505  ovolfsf  25506  ovollb2lem  25523  ovolshftlem1  25544  ovolscalem1  25548  ioombl1lem1  25593  ioombl1lem3  25595  ioombl1lem4  25596  ovolfs2  25606  uniiccdif  25613  uniioovol  25614  uniioombllem2a  25617  uniioombllem2  25618  uniioombllem3a  25619  uniioombllem3  25620  uniioombllem4  25621  uniioombllem6  25623  ovolval3  46662
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