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Theorem ovolfcl 24983
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 7084 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
21elin2d 4200 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ (ℝ × ℝ))
3 1st2nd2 8014 . . . 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 2835 . 2 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
6 ancom 462 . . 3 (((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ∧ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ)) ↔ (((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ) ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
7 elin 3965 . . . 4 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ ∧ ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ (ℝ × ℝ)))
8 df-br 5150 . . . . . 6 ((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ↔ ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ )
98bicomi 223 . . . . 5 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ ↔ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)))
10 opelxp 5713 . . . . 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 1090 . . 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 217 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 397  w3a 1088   = wceq 1542  wcel 2107  cin 3948  cop 4635   class class class wbr 5149   × cxp 5675  wf 6540  cfv 6544  1st c1st 7973  2nd c2nd 7974  cr 11109  cle 11249  cn 12212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-1st 7975  df-2nd 7976
This theorem is referenced by:  ovolfioo  24984  ovolficc  24985  ovolfsval  24987  ovolfsf  24988  ovollb2lem  25005  ovolshftlem1  25026  ovolscalem1  25030  ioombl1lem1  25075  ioombl1lem3  25077  ioombl1lem4  25078  ovolfs2  25088  uniiccdif  25095  uniioovol  25096  uniioombllem2a  25099  uniioombllem2  25100  uniioombllem3a  25101  uniioombllem3  25102  uniioombllem4  25103  uniioombllem6  25105  ovolval3  45363
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