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Theorem ovolfcl 23454
 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 inss2 3982 . . . . 5 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
2 ffvelrn 6500 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
31, 2sseldi 3750 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ (ℝ × ℝ))
4 1st2nd2 7354 . . . 4 ((𝐹𝑁) ∈ (ℝ × ℝ) → (𝐹𝑁) = ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩)
53, 4syl 17 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) = ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩)
65, 2eqeltrrd 2851 . 2 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
7 ancom 452 . . 3 (((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ∧ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ)) ↔ (((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ) ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
8 elin 3947 . . . 4 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ ∧ ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ (ℝ × ℝ)))
9 df-br 4787 . . . . . 6 ((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ↔ ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ )
109bicomi 214 . . . . 5 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ ↔ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)))
11 opelxp 5286 . . . . 5 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ (ℝ × ℝ) ↔ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ))
1210, 11anbi12i 612 . . . 4 ((⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ≤ ∧ ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ (ℝ × ℝ)) ↔ ((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ∧ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ)))
138, 12bitri 264 . . 3 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ ((1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁)) ∧ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ)))
14 df-3an 1073 . . 3 (((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))) ↔ (((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ) ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
157, 13, 143bitr4i 292 . 2 (⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
166, 15sylib 208 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 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ∩ cin 3722  ⟨cop 4322   class class class wbr 4786   × cxp 5247  ⟶wf 6027  ‘cfv 6031  1st c1st 7313  2nd c2nd 7314  ℝcr 10137   ≤ cle 10277  ℕcn 11222 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-1st 7315  df-2nd 7316 This theorem is referenced by:  ovolfioo  23455  ovolficc  23456  ovolfsval  23458  ovolfsf  23459  ovollb2lem  23476  ovolshftlem1  23497  ovolscalem1  23501  ioombl1lem1  23546  ioombl1lem3  23548  ioombl1lem4  23549  ovolfs2  23559  uniiccdif  23566  uniioovol  23567  uniioombllem2a  23570  uniioombllem2  23571  uniioombllem3a  23572  uniioombllem3  23573  uniioombllem4  23574  uniioombllem6  23576  ovolval3  41381
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