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Theorem fvinim0ffz 10140
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 5319 . . . . . 6 (𝐹:(0...𝐾)⟶𝑉𝐹 Fn (0...𝐾))
21adantr 274 . . . . 5 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → 𝐹 Fn (0...𝐾))
3 0nn0 9105 . . . . . . 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 9115 . . . . . . 7 (𝐾 ∈ ℕ0 → 0 ≤ 𝐾)
76adantl 275 . . . . . 6 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → 0 ≤ 𝐾)
8 elfz2nn0 10014 . . . . . 6 (0 ∈ (0...𝐾) ↔ (0 ∈ ℕ0𝐾 ∈ ℕ0 ∧ 0 ≤ 𝐾))
94, 5, 7, 8syl3anbrc 1166 . . . . 5 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → 0 ∈ (0...𝐾))
10 id 19 . . . . . . 7 (𝐾 ∈ ℕ0𝐾 ∈ ℕ0)
11 nn0re 9099 . . . . . . . 8 (𝐾 ∈ ℕ0𝐾 ∈ ℝ)
1211leidd 8389 . . . . . . 7 (𝐾 ∈ ℕ0𝐾𝐾)
13 elfz2nn0 10014 . . . . . . 7 (𝐾 ∈ (0...𝐾) ↔ (𝐾 ∈ ℕ0𝐾 ∈ ℕ0𝐾𝐾))
1410, 10, 12, 13syl3anbrc 1166 . . . . . 6 (𝐾 ∈ ℕ0𝐾 ∈ (0...𝐾))
1514adantl 275 . . . . 5 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → 𝐾 ∈ (0...𝐾))
16 fnimapr 5528 . . . . 5 ((𝐹 Fn (0...𝐾) ∧ 0 ∈ (0...𝐾) ∧ 𝐾 ∈ (0...𝐾)) → (𝐹 “ {0, 𝐾}) = {(𝐹‘0), (𝐹𝐾)})
172, 9, 15, 16syl3anc 1220 . . . 4 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝐹 “ {0, 𝐾}) = {(𝐹‘0), (𝐹𝐾)})
1817ineq1d 3307 . . 3 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ({(𝐹‘0), (𝐹𝐾)} ∩ (𝐹 “ (1..^𝐾))))
1918eqeq1d 2166 . 2 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ({(𝐹‘0), (𝐹𝐾)} ∩ (𝐹 “ (1..^𝐾))) = ∅))
20 disj 3442 . . 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 5603 . . . 4 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝐹‘0) ∈ 𝑉)
2321, 15ffvelrnd 5603 . . . 4 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (𝐹𝐾) ∈ 𝑉)
24 eleq1 2220 . . . . . . 7 (𝑣 = (𝐹‘0) → (𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾))))
2524notbid 657 . . . . . 6 (𝑣 = (𝐹‘0) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾))))
26 df-nel 2423 . . . . . 6 ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
2725, 26bitr4di 197 . . . . 5 (𝑣 = (𝐹‘0) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘0) ∉ (𝐹 “ (1..^𝐾))))
28 eleq1 2220 . . . . . . 7 (𝑣 = (𝐹𝐾) → (𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾))))
2928notbid 657 . . . . . 6 (𝑣 = (𝐹𝐾) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾))))
30 df-nel 2423 . . . . . 6 ((𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹𝐾) ∈ (𝐹 “ (1..^𝐾)))
3129, 30bitr4di 197 . . . . 5 (𝑣 = (𝐹𝐾) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾))))
3227, 31ralprg 3610 . . . 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 409 . . 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 1335  wcel 2128  wnel 2422  wral 2435  cin 3101  c0 3394  {cpr 3561   class class class wbr 3965  cima 4589   Fn wfn 5165  wf 5166  cfv 5170  (class class class)co 5824  0cc0 7732  1c1 7733  cle 7913  0cn0 9090  ...cfz 9912  ..^cfzo 10041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-ltadd 7848
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-inn 8834  df-n0 9091  df-z 9168  df-uz 9440  df-fz 9913
This theorem is referenced by: (None)
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