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Theorem ovolfcl 25515
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 4215 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ (ℝ × ℝ))
3 1st2nd2 8052 . . . 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 2840 . 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 3979 . . . 4 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ ∧ ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ (ℝ × ℝ)))
8 df-br 5149 . . . . . 6 ((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ↔ ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ )
98bicomi 224 . . . . 5 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ ↔ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)))
10 opelxp 5725 . . . . 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 1088 . . 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 1086   = wceq 1537  wcel 2106  cin 3962  cop 4637   class class class wbr 5148   × cxp 5687  wf 6559  cfv 6563  1st c1st 8011  2nd c2nd 8012  cr 11152  cle 11294  cn 12264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  ovolfioo  25516  ovolficc  25517  ovolfsval  25519  ovolfsf  25520  ovollb2lem  25537  ovolshftlem1  25558  ovolscalem1  25562  ioombl1lem1  25607  ioombl1lem3  25609  ioombl1lem4  25610  ovolfs2  25620  uniiccdif  25627  uniioovol  25628  uniioombllem2a  25631  uniioombllem2  25632  uniioombllem3a  25633  uniioombllem3  25634  uniioombllem4  25635  uniioombllem6  25637  ovolval3  46603
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