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Theorem fvinim0ffz 9905
Description: The function values for the borders of a finite interval of integers, which is the domain of the function, are not in the image of the interior of the interval iff the intersection of the images of the interior and the borders is empty. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 5-Feb-2021.)
Assertion
Ref Expression
fvinim0ffz ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))

Proof of Theorem fvinim0ffz
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 ffn 5228 . . . . . 6 (𝐹:(0...𝐾)⟶𝑉𝐹 Fn (0...𝐾))
21adantr 272 . . . . 5 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → 𝐹 Fn (0...𝐾))
3 0nn0 8890 . . . . . . 7 0 ∈ ℕ0
43a1i 9 . . . . . 6 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → 0 ∈ ℕ0)
5 simpr 109 . . . . . 6 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
6 nn0ge0 8900 . . . . . . 7 (𝐾 ∈ ℕ0 → 0 ≤ 𝐾)
76adantl 273 . . . . . 6 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → 0 ≤ 𝐾)
8 elfz2nn0 9779 . . . . . 6 (0 ∈ (0...𝐾) ↔ (0 ∈ ℕ0𝐾 ∈ ℕ0 ∧ 0 ≤ 𝐾))
94, 5, 7, 8syl3anbrc 1146 . . . . 5 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → 0 ∈ (0...𝐾))
10 id 19 . . . . . . 7 (𝐾 ∈ ℕ0𝐾 ∈ ℕ0)
11 nn0re 8884 . . . . . . . 8 (𝐾 ∈ ℕ0𝐾 ∈ ℝ)
1211leidd 8189 . . . . . . 7 (𝐾 ∈ ℕ0𝐾𝐾)
13 elfz2nn0 9779 . . . . . . 7 (𝐾 ∈ (0...𝐾) ↔ (𝐾 ∈ ℕ0𝐾 ∈ ℕ0𝐾𝐾))
1410, 10, 12, 13syl3anbrc 1146 . . . . . 6 (𝐾 ∈ ℕ0𝐾 ∈ (0...𝐾))
1514adantl 273 . . . . 5 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → 𝐾 ∈ (0...𝐾))
16 fnimapr 5433 . . . . 5 ((𝐹 Fn (0...𝐾) ∧ 0 ∈ (0...𝐾) ∧ 𝐾 ∈ (0...𝐾)) → (𝐹 “ {0, 𝐾}) = {(𝐹‘0), (𝐹𝐾)})
172, 9, 15, 16syl3anc 1197 . . . 4 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝐹 “ {0, 𝐾}) = {(𝐹‘0), (𝐹𝐾)})
1817ineq1d 3240 . . 3 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ({(𝐹‘0), (𝐹𝐾)} ∩ (𝐹 “ (1..^𝐾))))
1918eqeq1d 2121 . 2 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ({(𝐹‘0), (𝐹𝐾)} ∩ (𝐹 “ (1..^𝐾))) = ∅))
20 disj 3375 . . 3 (({(𝐹‘0), (𝐹𝐾)} ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ∀𝑣 ∈ {(𝐹‘0), (𝐹𝐾)} ¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)))
21 simpl 108 . . . . 5 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → 𝐹:(0...𝐾)⟶𝑉)
2221, 9ffvelrnd 5508 . . . 4 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝐹‘0) ∈ 𝑉)
2321, 15ffvelrnd 5508 . . . 4 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝐹𝐾) ∈ 𝑉)
24 eleq1 2175 . . . . . . 7 (𝑣 = (𝐹‘0) → (𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾))))
2524notbid 639 . . . . . 6 (𝑣 = (𝐹‘0) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾))))
26 df-nel 2376 . . . . . 6 ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
2725, 26syl6bbr 197 . . . . 5 (𝑣 = (𝐹‘0) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘0) ∉ (𝐹 “ (1..^𝐾))))
28 eleq1 2175 . . . . . . 7 (𝑣 = (𝐹𝐾) → (𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾))))
2928notbid 639 . . . . . 6 (𝑣 = (𝐹𝐾) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾))))
30 df-nel 2376 . . . . . 6 ((𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)))
3129, 30syl6bbr 197 . . . . 5 (𝑣 = (𝐹𝐾) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾))))
3227, 31ralprg 3538 . . . 4 (((𝐹‘0) ∈ 𝑉 ∧ (𝐹𝐾) ∈ 𝑉) → (∀𝑣 ∈ {(𝐹‘0), (𝐹𝐾)} ¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
3322, 23, 32syl2anc 406 . . 3 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (∀𝑣 ∈ {(𝐹‘0), (𝐹𝐾)} ¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
3420, 33syl5bb 191 . 2 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (({(𝐹‘0), (𝐹𝐾)} ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
3519, 34bitrd 187 1 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104   = wceq 1312  wcel 1461  wnel 2375  wral 2388  cin 3034  c0 3327  {cpr 3492   class class class wbr 3893  cima 4500   Fn wfn 5074  wf 5075  cfv 5079  (class class class)co 5726  0cc0 7541  1c1 7542  cle 7719  0cn0 8875  ...cfz 9677  ..^cfzo 9806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-addcom 7639  ax-addass 7641  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-0id 7647  ax-rnegex 7648  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-ltadd 7655
This theorem depends on definitions:  df-bi 116  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-inn 8625  df-n0 8876  df-z 8953  df-uz 9223  df-fz 9678
This theorem is referenced by: (None)
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