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Mirrors > Home > MPE Home > Th. List > fvinim0ffz | Structured version Visualization version GIF version |
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.) |
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))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6513 | . . . . . 6 ⊢ (𝐹:(0...𝐾)⟶𝑉 → 𝐹 Fn (0...𝐾)) | |
2 | 1 | adantr 483 | . . . . 5 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → 𝐹 Fn (0...𝐾)) |
3 | 0nn0 11911 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
4 | 3 | a1i 11 | . . . . . 6 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → 0 ∈ ℕ0) |
5 | simpr 487 | . . . . . 6 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0) | |
6 | nn0ge0 11921 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ0 → 0 ≤ 𝐾) | |
7 | 6 | adantl 484 | . . . . . 6 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → 0 ≤ 𝐾) |
8 | elfz2nn0 12997 | . . . . . 6 ⊢ (0 ∈ (0...𝐾) ↔ (0 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ∧ 0 ≤ 𝐾)) | |
9 | 4, 5, 7, 8 | syl3anbrc 1339 | . . . . 5 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → 0 ∈ (0...𝐾)) |
10 | id 22 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ0 → 𝐾 ∈ ℕ0) | |
11 | nn0re 11905 | . . . . . . . 8 ⊢ (𝐾 ∈ ℕ0 → 𝐾 ∈ ℝ) | |
12 | 11 | leidd 11205 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ0 → 𝐾 ≤ 𝐾) |
13 | elfz2nn0 12997 | . . . . . . 7 ⊢ (𝐾 ∈ (0...𝐾) ↔ (𝐾 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝐾)) | |
14 | 10, 10, 12, 13 | syl3anbrc 1339 | . . . . . 6 ⊢ (𝐾 ∈ ℕ0 → 𝐾 ∈ (0...𝐾)) |
15 | 14 | adantl 484 | . . . . 5 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ (0...𝐾)) |
16 | fnimapr 6746 | . . . . 5 ⊢ ((𝐹 Fn (0...𝐾) ∧ 0 ∈ (0...𝐾) ∧ 𝐾 ∈ (0...𝐾)) → (𝐹 “ {0, 𝐾}) = {(𝐹‘0), (𝐹‘𝐾)}) | |
17 | 2, 9, 15, 16 | syl3anc 1367 | . . . 4 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (𝐹 “ {0, 𝐾}) = {(𝐹‘0), (𝐹‘𝐾)}) |
18 | 17 | ineq1d 4187 | . . 3 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ({(𝐹‘0), (𝐹‘𝐾)} ∩ (𝐹 “ (1..^𝐾)))) |
19 | 18 | eqeq1d 2823 | . 2 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ({(𝐹‘0), (𝐹‘𝐾)} ∩ (𝐹 “ (1..^𝐾))) = ∅)) |
20 | disj 4398 | . . 3 ⊢ (({(𝐹‘0), (𝐹‘𝐾)} ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ∀𝑣 ∈ {(𝐹‘0), (𝐹‘𝐾)} ¬ 𝑣 ∈ (𝐹 “ (1..^𝐾))) | |
21 | fvex 6682 | . . . 4 ⊢ (𝐹‘0) ∈ V | |
22 | fvex 6682 | . . . 4 ⊢ (𝐹‘𝐾) ∈ V | |
23 | eleq1 2900 | . . . . . 6 ⊢ (𝑣 = (𝐹‘0) → (𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))) | |
24 | 23 | notbid 320 | . . . . 5 ⊢ (𝑣 = (𝐹‘0) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾)))) |
25 | df-nel 3124 | . . . . 5 ⊢ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘0) ∈ (𝐹 “ (1..^𝐾))) | |
26 | 24, 25 | syl6bbr 291 | . . . 4 ⊢ (𝑣 = (𝐹‘0) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘0) ∉ (𝐹 “ (1..^𝐾)))) |
27 | eleq1 2900 | . . . . . 6 ⊢ (𝑣 = (𝐹‘𝐾) → (𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)))) | |
28 | 27 | notbid 320 | . . . . 5 ⊢ (𝑣 = (𝐹‘𝐾) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾)))) |
29 | df-nel 3124 | . . . . 5 ⊢ ((𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)) ↔ ¬ (𝐹‘𝐾) ∈ (𝐹 “ (1..^𝐾))) | |
30 | 28, 29 | syl6bbr 291 | . . . 4 ⊢ (𝑣 = (𝐹‘𝐾) → (¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)))) |
31 | 21, 22, 26, 30 | ralpr 4635 | . . 3 ⊢ (∀𝑣 ∈ {(𝐹‘0), (𝐹‘𝐾)} ¬ 𝑣 ∈ (𝐹 “ (1..^𝐾)) ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)))) |
32 | 20, 31 | bitri 277 | . 2 ⊢ (({(𝐹‘0), (𝐹‘𝐾)} ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)))) |
33 | 19, 32 | syl6bb 289 | 1 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾))))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∉ wnel 3123 ∀wral 3138 ∩ cin 3934 ∅c0 4290 {cpr 4568 class class class wbr 5065 “ cima 5557 Fn wfn 6349 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 0cc0 10536 1c1 10537 ≤ cle 10675 ℕ0cn0 11896 ...cfz 12891 ..^cfzo 13032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 |
This theorem is referenced by: injresinjlem 13156 pthdivtx 27509 pthdlem2 27548 |
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