Theorem fvinim0ffz 10227

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 5361	. . . . . 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 9180	. . . . . . 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 9190	. . . . . . 7 ⊢ (𝐾 ∈ ℕ0 → 0 ≤ 𝐾)
7	6	adantl 277	. . . . . 6 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → 0 ≤ 𝐾)
8		elfz2nn0 10098	. . . . . 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 9174	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ0 → 𝐾 ∈ ℝ)
12	11	leidd 8461	. . . . . . 7 ⊢ (𝐾 ∈ ℕ0 → 𝐾 ≤ 𝐾)
13		elfz2nn0 10098	. . . . . . 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 5572	. . . . 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 3335	. . 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 3471	. . 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 5648	. . . 4 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (𝐹‘0) ∈ 𝑉)
23	21, 15	ffvelcdmd 5648	. . . 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 3642	. . . 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 3128  ∅c0 3422  {cpr 3592   class class class wbr 4000   “ cima 4626   Fn wfn 5207  ⟶wf 5208  ‘cfv 5212  (class class class)co 5869  0cc0 7802  1c1 7803   ≤ cle 7983  ℕ₀cn0 9165  ...cfz 9995  ..^cfzo 10128

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 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918

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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996

This theorem is referenced by: (None)