Theorem fvinim0ffz 10241

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)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)))))

Proof of Theorem fvinim0ffz

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 5366	. . . . . 6 ⊢ (𝐹:(0...𝐾)⟶𝑉 → 𝐹 Fn (0...𝐾))
2	1	adantr 276	. . . . 5 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → 𝐹 Fn (0...𝐾))
3		0nn0 9191	. . . . . . 7 ⊢ 0 ∈ ℕ0
4	3	a1i 9	. . . . . 6 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → 0 ∈ ℕ0)
5		simpr 110	. . . . . 6 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
6		nn0ge0 9201	. . . . . . 7 ⊢ (𝐾 ∈ ℕ0 → 0 ≤ 𝐾)
7	6	adantl 277	. . . . . 6 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → 0 ≤ 𝐾)
8		elfz2nn0 10112	. . . . . 6 ⊢ (0 ∈ (0...𝐾) ↔ (0 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ∧ 0 ≤ 𝐾))
9	4, 5, 7, 8	syl3anbrc 1181	. . . . 5 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → 0 ∈ (0...𝐾))
10		id 19	. . . . . . 7 ⊢ (𝐾 ∈ ℕ0 → 𝐾 ∈ ℕ0)
11		nn0re 9185	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ0 → 𝐾 ∈ ℝ)
12	11	leidd 8471	. . . . . . 7 ⊢ (𝐾 ∈ ℕ0 → 𝐾 ≤ 𝐾)
13		elfz2nn0 10112	. . . . . . 7 ⊢ (𝐾 ∈ (0...𝐾) ↔ (𝐾 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝐾))
14	10, 10, 12, 13	syl3anbrc 1181	. . . . . 6 ⊢ (𝐾 ∈ ℕ0 → 𝐾 ∈ (0...𝐾))
15	14	adantl 277	. . . . 5 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ (0...𝐾))
16		fnimapr 5577	. . . . 5 ⊢ ((𝐹 Fn (0...𝐾) ∧ 0 ∈ (0...𝐾) ∧ 𝐾 ∈ (0...𝐾)) → (𝐹 “ {0, 𝐾}) = {(𝐹‘0), (𝐹‘𝐾)})
17	2, 9, 15, 16	syl3anc 1238	. . . 4 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (𝐹 “ {0, 𝐾}) = {(𝐹‘0), (𝐹‘𝐾)})
18	17	ineq1d 3336	. . 3 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ({(𝐹‘0), (𝐹‘𝐾)} ∩ (𝐹 “ (1..^𝐾))))
19	18	eqeq1d 2186	. 2 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ({(𝐹‘0), (𝐹‘𝐾)} ∩ (𝐹 “ (1..^𝐾))) = ∅))
20		disj 3472	. . 3 ⊢ (({(𝐹‘0), (𝐹‘𝐾)} ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ∀𝑣 ∈ {(𝐹‘0), (𝐹‘𝐾)} ¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)))
21		simpl 109	. . . . 5 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → 𝐹:(0...𝐾)⟶𝑉)
22	21, 9	ffvelcdmd 5653	. . . 4 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (𝐹‘0) ∈ 𝑉)
23	21, 15	ffvelcdmd 5653	. . . 4 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (𝐹‘𝐾) ∈ 𝑉)
24		eleq1 2240	. . . . . . 7 ⊢ (𝑣 = (𝐹‘0) → (𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾))))
25	24	notbid 667	. . . . . 6 ⊢ (𝑣 = (𝐹‘0) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾))))
26		df-nel 2443	. . . . . 6 ⊢ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))
27	25, 26	bitr4di 198	. . . . 5 ⊢ (𝑣 = (𝐹‘0) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘0) ∉ (𝐹 “ (1..^𝐾))))
28		eleq1 2240	. . . . . . 7 ⊢ (𝑣 = (𝐹‘𝐾) → (𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾))))
29	28	notbid 667	. . . . . 6 ⊢ (𝑣 = (𝐹‘𝐾) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾))))
30		df-nel 2443	. . . . . 6 ⊢ ((𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)))
31	29, 30	bitr4di 198	. . . . 5 ⊢ (𝑣 = (𝐹‘𝐾) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾))))
32	27, 31	ralprg 3644	. . . 4 ⊢ (((𝐹‘0) ∈ 𝑉 ∧ (𝐹‘𝐾) ∈ 𝑉) → (∀𝑣 ∈ {(𝐹‘0), (𝐹‘𝐾)} ¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)))))
33	22, 23, 32	syl2anc 411	. . 3 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (∀𝑣 ∈ {(𝐹‘0), (𝐹‘𝐾)} ¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)))))
34	20, 33	bitrid 192	. 2 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (({(𝐹‘0), (𝐹‘𝐾)} ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)))))
35	19, 34	bitrd 188	1 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148   ∉ wnel 2442  ∀wral 2455   ∩ cin 3129  ∅c0 3423  {cpr 3594   class class class wbr 4004   “ cima 4630   Fn wfn 5212  ⟶wf 5213  ‘cfv 5217  (class class class)co 5875  0cc0 7811  1c1 7812   ≤ cle 7993  ℕ₀cn0 9176  ...cfz 10008  ..^cfzo 10142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529  df-fz 10009

This theorem is referenced by: (None)