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Theorem f1resfz0f1d 35098
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 13660 . . 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 13661 . . . . . 6 (𝐾 ∈ ℕ0 → 0 ∈ (0...𝐾))
7 snssi 4813 . . . . . 6 (0 ∈ (0...𝐾) → {0} ⊆ (0...𝐾))
85, 6, 73syl 18 . . . . 5 (𝜑 → {0} ⊆ (0...𝐾))
93, 8fssresd 6776 . . . 4 (𝜑 → (𝐹 ↾ {0}):{0}⟶𝑉)
10 eqidd 2736 . . . . 5 (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0)
11 0nn0 12539 . . . . . 6 0 ∈ ℕ0
12 fveqeq2 6916 . . . . . . . 8 (𝑥 = 0 → (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦)))
13 eqeq1 2739 . . . . . . . 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 6907 . . . . . . . . 9 (𝑦 = 0 → ((𝐹 ↾ {0})‘𝑦) = ((𝐹 ↾ {0})‘0))
1615eqeq2d 2746 . . . . . . . 8 (𝑦 = 0 → (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0)))
17 eqeq2 2747 . . . . . . . 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 4683 . . . . . 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 4168 . . . . . . . 8 ((1...𝐾) ∪ {0}) = ({0} ∪ (1...𝐾))
25 fz0sn0fz1 13682 . . . . . . . . 9 (𝐾 ∈ ℕ0 → (0...𝐾) = ({0} ∪ (1...𝐾)))
265, 25syl 17 . . . . . . . 8 (𝜑 → (0...𝐾) = ({0} ∪ (1...𝐾)))
2724, 26eqtr4id 2794 . . . . . . 7 (𝜑 → ((1...𝐾) ∪ {0}) = (0...𝐾))
28 0nelfz1 13580 . . . . . . . . . 10 0 ∉ (1...𝐾)
2928neli 3046 . . . . . . . . 9 ¬ 0 ∈ (1...𝐾)
30 disjsn 4716 . . . . . . . . 9 (((1...𝐾) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...𝐾))
3129, 30mpbir 231 . . . . . . . 8 ((1...𝐾) ∩ {0}) = ∅
32 uneqdifeq 4499 . . . . . . . 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 2741 . . . . 5 (𝜑 → {0} = ((0...𝐾) ∖ (1...𝐾)))
3635reseq2d 6000 . . . 4 (𝜑 → (𝐹 ↾ {0}) = (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))))
37 eqidd 2736 . . . 4 (𝜑𝑉 = 𝑉)
3836, 35, 37f1eq123d 6841 . . 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 6080 . . . 4 (𝜑 → (𝐹 “ {0}) = (𝐹 “ ((0...𝐾) ∖ (1...𝐾))))
4140ineq2d 4228 . . 3 (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))))
42 incom 4217 . . . 4 ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0}))
43 f1resfz0f1d.4 . . . 4 (𝜑 → ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ∅)
4442, 43eqtr3id 2789 . . 3 (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ∅)
4541, 44eqtr3d 2777 . 2 (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))) = ∅)
462, 3, 4, 39, 45f1resrcmplf1d 35080 1 (𝜑𝐹:(0...𝐾)–1-1𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1537  wcel 2106  wral 3059  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  {csn 4631  cres 5691  cima 5692  wf 6559  1-1wf1 6560  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154  0cn0 12524  ...cfz 13544
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545
This theorem is referenced by:  pthhashvtx  35112
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