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Theorem f1resfz0f1d 35119
Description: If a function with a sequence of nonnegative integers (starting at 0) as its domain is one-to-one when 0 is removed, and if the range of that restriction does not contain the function's value at the removed integer, then the function is itself one-to-one. (Contributed by BTernaryTau, 4-Oct-2023.)
Hypotheses
Ref Expression
f1resfz0f1d.1 (𝜑𝐾 ∈ ℕ0)
f1resfz0f1d.2 (𝜑𝐹:(0...𝐾)⟶𝑉)
f1resfz0f1d.3 (𝜑 → (𝐹 ↾ (1...𝐾)):(1...𝐾)–1-1𝑉)
f1resfz0f1d.4 (𝜑 → ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ∅)
Assertion
Ref Expression
f1resfz0f1d (𝜑𝐹:(0...𝐾)–1-1𝑉)

Proof of Theorem f1resfz0f1d
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fz1ssfz0 13663 . . 3 (1...𝐾) ⊆ (0...𝐾)
21a1i 11 . 2 (𝜑 → (1...𝐾) ⊆ (0...𝐾))
3 f1resfz0f1d.2 . 2 (𝜑𝐹:(0...𝐾)⟶𝑉)
4 f1resfz0f1d.3 . 2 (𝜑 → (𝐹 ↾ (1...𝐾)):(1...𝐾)–1-1𝑉)
5 f1resfz0f1d.1 . . . . . 6 (𝜑𝐾 ∈ ℕ0)
6 0elfz 13664 . . . . . 6 (𝐾 ∈ ℕ0 → 0 ∈ (0...𝐾))
7 snssi 4808 . . . . . 6 (0 ∈ (0...𝐾) → {0} ⊆ (0...𝐾))
85, 6, 73syl 18 . . . . 5 (𝜑 → {0} ⊆ (0...𝐾))
93, 8fssresd 6775 . . . 4 (𝜑 → (𝐹 ↾ {0}):{0}⟶𝑉)
10 eqidd 2738 . . . . 5 (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0)
11 0nn0 12541 . . . . . 6 0 ∈ ℕ0
12 fveqeq2 6915 . . . . . . . 8 (𝑥 = 0 → (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦)))
13 eqeq1 2741 . . . . . . . 8 (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
1412, 13imbi12d 344 . . . . . . 7 (𝑥 = 0 → ((((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) → 0 = 𝑦)))
15 fveq2 6906 . . . . . . . . 9 (𝑦 = 0 → ((𝐹 ↾ {0})‘𝑦) = ((𝐹 ↾ {0})‘0))
1615eqeq2d 2748 . . . . . . . 8 (𝑦 = 0 → (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0)))
17 eqeq2 2749 . . . . . . . 8 (𝑦 = 0 → (0 = 𝑦 ↔ 0 = 0))
1816, 17imbi12d 344 . . . . . . 7 (𝑦 = 0 → ((((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) → 0 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0)))
1914, 182ralsng 4678 . . . . . 6 ((0 ∈ ℕ0 ∧ 0 ∈ ℕ0) → (∀𝑥 ∈ {0}∀𝑦 ∈ {0} (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0)))
2011, 11, 19mp2an 692 . . . . 5 (∀𝑥 ∈ {0}∀𝑦 ∈ {0} (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0))
2110, 20mpbir 231 . . . 4 𝑥 ∈ {0}∀𝑦 ∈ {0} (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦)
22 dff13 7275 . . . 4 ((𝐹 ↾ {0}):{0}–1-1𝑉 ↔ ((𝐹 ↾ {0}):{0}⟶𝑉 ∧ ∀𝑥 ∈ {0}∀𝑦 ∈ {0} (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦)))
239, 21, 22sylanblrc 590 . . 3 (𝜑 → (𝐹 ↾ {0}):{0}–1-1𝑉)
24 uncom 4158 . . . . . . . 8 ((1...𝐾) ∪ {0}) = ({0} ∪ (1...𝐾))
25 fz0sn0fz1 13685 . . . . . . . . 9 (𝐾 ∈ ℕ0 → (0...𝐾) = ({0} ∪ (1...𝐾)))
265, 25syl 17 . . . . . . . 8 (𝜑 → (0...𝐾) = ({0} ∪ (1...𝐾)))
2724, 26eqtr4id 2796 . . . . . . 7 (𝜑 → ((1...𝐾) ∪ {0}) = (0...𝐾))
28 0nelfz1 13583 . . . . . . . . . 10 0 ∉ (1...𝐾)
2928neli 3048 . . . . . . . . 9 ¬ 0 ∈ (1...𝐾)
30 disjsn 4711 . . . . . . . . 9 (((1...𝐾) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...𝐾))
3129, 30mpbir 231 . . . . . . . 8 ((1...𝐾) ∩ {0}) = ∅
32 uneqdifeq 4493 . . . . . . . 8 (((1...𝐾) ⊆ (0...𝐾) ∧ ((1...𝐾) ∩ {0}) = ∅) → (((1...𝐾) ∪ {0}) = (0...𝐾) ↔ ((0...𝐾) ∖ (1...𝐾)) = {0}))
331, 31, 32mp2an 692 . . . . . . 7 (((1...𝐾) ∪ {0}) = (0...𝐾) ↔ ((0...𝐾) ∖ (1...𝐾)) = {0})
3427, 33sylib 218 . . . . . 6 (𝜑 → ((0...𝐾) ∖ (1...𝐾)) = {0})
3534eqcomd 2743 . . . . 5 (𝜑 → {0} = ((0...𝐾) ∖ (1...𝐾)))
3635reseq2d 5997 . . . 4 (𝜑 → (𝐹 ↾ {0}) = (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))))
37 eqidd 2738 . . . 4 (𝜑𝑉 = 𝑉)
3836, 35, 37f1eq123d 6840 . . 3 (𝜑 → ((𝐹 ↾ {0}):{0}–1-1𝑉 ↔ (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))):((0...𝐾) ∖ (1...𝐾))–1-1𝑉))
3923, 38mpbid 232 . 2 (𝜑 → (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))):((0...𝐾) ∖ (1...𝐾))–1-1𝑉)
4035imaeq2d 6078 . . . 4 (𝜑 → (𝐹 “ {0}) = (𝐹 “ ((0...𝐾) ∖ (1...𝐾))))
4140ineq2d 4220 . . 3 (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))))
42 incom 4209 . . . 4 ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0}))
43 f1resfz0f1d.4 . . . 4 (𝜑 → ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ∅)
4442, 43eqtr3id 2791 . . 3 (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ∅)
4541, 44eqtr3d 2779 . 2 (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))) = ∅)
462, 3, 4, 39, 45f1resrcmplf1d 35101 1 (𝜑𝐹:(0...𝐾)–1-1𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1540  wcel 2108  wral 3061  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626  cres 5687  cima 5688  wf 6557  1-1wf1 6558  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  0cn0 12526  ...cfz 13547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548
This theorem is referenced by:  pthhashvtx  35133
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