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Theorem f1resfz0f1d 32354
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 12995 . . 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 12996 . . . . . 6 (𝐾 ∈ ℕ0 → 0 ∈ (0...𝐾))
7 snssi 4733 . . . . . 6 (0 ∈ (0...𝐾) → {0} ⊆ (0...𝐾))
85, 6, 73syl 18 . . . . 5 (𝜑 → {0} ⊆ (0...𝐾))
93, 8fssresd 6538 . . . 4 (𝜑 → (𝐹 ↾ {0}):{0}⟶𝑉)
10 eqidd 2820 . . . . 5 (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0)
11 0nn0 11904 . . . . . 6 0 ∈ ℕ0
12 fveqeq2 6672 . . . . . . . 8 (𝑥 = 0 → (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦)))
13 eqeq1 2823 . . . . . . . 8 (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
1412, 13imbi12d 347 . . . . . . 7 (𝑥 = 0 → ((((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) → 0 = 𝑦)))
15 fveq2 6663 . . . . . . . . 9 (𝑦 = 0 → ((𝐹 ↾ {0})‘𝑦) = ((𝐹 ↾ {0})‘0))
1615eqeq2d 2830 . . . . . . . 8 (𝑦 = 0 → (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0)))
17 eqeq2 2831 . . . . . . . 8 (𝑦 = 0 → (0 = 𝑦 ↔ 0 = 0))
1816, 17imbi12d 347 . . . . . . 7 (𝑦 = 0 → ((((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) → 0 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0)))
1914, 182ralsng 4608 . . . . . 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 690 . . . . 5 (∀𝑥 ∈ {0}∀𝑦 ∈ {0} (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0))
2110, 20mpbir 233 . . . 4 𝑥 ∈ {0}∀𝑦 ∈ {0} (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦)
22 dff13 7005 . . . 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 592 . . 3 (𝜑 → (𝐹 ↾ {0}):{0}–1-1𝑉)
24 fz0sn0fz1 13016 . . . . . . . . 9 (𝐾 ∈ ℕ0 → (0...𝐾) = ({0} ∪ (1...𝐾)))
255, 24syl 17 . . . . . . . 8 (𝜑 → (0...𝐾) = ({0} ∪ (1...𝐾)))
26 uncom 4127 . . . . . . . 8 ((1...𝐾) ∪ {0}) = ({0} ∪ (1...𝐾))
2725, 26syl6reqr 2873 . . . . . . 7 (𝜑 → ((1...𝐾) ∪ {0}) = (0...𝐾))
28 0nelfz1 12918 . . . . . . . . . 10 0 ∉ (1...𝐾)
2928neli 3123 . . . . . . . . 9 ¬ 0 ∈ (1...𝐾)
30 disjsn 4639 . . . . . . . . 9 (((1...𝐾) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...𝐾))
3129, 30mpbir 233 . . . . . . . 8 ((1...𝐾) ∩ {0}) = ∅
32 uneqdifeq 4436 . . . . . . . 8 (((1...𝐾) ⊆ (0...𝐾) ∧ ((1...𝐾) ∩ {0}) = ∅) → (((1...𝐾) ∪ {0}) = (0...𝐾) ↔ ((0...𝐾) ∖ (1...𝐾)) = {0}))
331, 31, 32mp2an 690 . . . . . . 7 (((1...𝐾) ∪ {0}) = (0...𝐾) ↔ ((0...𝐾) ∖ (1...𝐾)) = {0})
3427, 33sylib 220 . . . . . 6 (𝜑 → ((0...𝐾) ∖ (1...𝐾)) = {0})
3534eqcomd 2825 . . . . 5 (𝜑 → {0} = ((0...𝐾) ∖ (1...𝐾)))
3635reseq2d 5846 . . . 4 (𝜑 → (𝐹 ↾ {0}) = (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))))
37 eqidd 2820 . . . 4 (𝜑𝑉 = 𝑉)
3836, 35, 37f1eq123d 6601 . . 3 (𝜑 → ((𝐹 ↾ {0}):{0}–1-1𝑉 ↔ (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))):((0...𝐾) ∖ (1...𝐾))–1-1𝑉))
3923, 38mpbid 234 . 2 (𝜑 → (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))):((0...𝐾) ∖ (1...𝐾))–1-1𝑉)
4035imaeq2d 5922 . . . 4 (𝜑 → (𝐹 “ {0}) = (𝐹 “ ((0...𝐾) ∖ (1...𝐾))))
4140ineq2d 4187 . . 3 (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))))
42 incom 4176 . . . 4 ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0}))
43 f1resfz0f1d.4 . . . 4 (𝜑 → ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ∅)
4442, 43syl5eqr 2868 . . 3 (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ∅)
4541, 44eqtr3d 2856 . 2 (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))) = ∅)
462, 3, 4, 39, 45f1resrcmplf1d 32353 1 (𝜑𝐹:(0...𝐾)–1-1𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208   = wceq 1531  wcel 2108  wral 3136  cdif 3931  cun 3932  cin 3933  wss 3934  c0 4289  {csn 4559  cres 5550  cima 5551  wf 6344  1-1wf1 6345  cfv 6348  (class class class)co 7148  0cc0 10529  1c1 10530  0cn0 11889  ...cfz 12884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885
This theorem is referenced by:  pthhashvtx  32367
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