Theorem f1resfz0f1d 33051

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	⊢ (𝜑 → 𝐾 ∈ ℕ0)
f1resfz0f1d.2	⊢ (𝜑 → 𝐹:(0...𝐾)⟶𝑉)
f1resfz0f1d.3	⊢ (𝜑 → (𝐹 ↾ (1...𝐾)):(1...𝐾)–1-1→𝑉)
f1resfz0f1d.4	⊢ (𝜑 → ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ∅)

Assertion

Ref	Expression
f1resfz0f1d	⊢ (𝜑 → 𝐹:(0...𝐾)–1-1→𝑉)

Proof of Theorem f1resfz0f1d

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fz1ssfz0 13334	. . 3 ⊢ (1...𝐾) ⊆ (0...𝐾)
2	1	a1i 11	. 2 ⊢ (𝜑 → (1...𝐾) ⊆ (0...𝐾))
3		f1resfz0f1d.2	. 2 ⊢ (𝜑 → 𝐹:(0...𝐾)⟶𝑉)
4		f1resfz0f1d.3	. 2 ⊢ (𝜑 → (𝐹 ↾ (1...𝐾)):(1...𝐾)–1-1→𝑉)
5		f1resfz0f1d.1	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
6		0elfz 13335	. . . . . 6 ⊢ (𝐾 ∈ ℕ0 → 0 ∈ (0...𝐾))
7		snssi 4746	. . . . . 6 ⊢ (0 ∈ (0...𝐾) → {0} ⊆ (0...𝐾))
8	5, 6, 7	3syl 18	. . . . 5 ⊢ (𝜑 → {0} ⊆ (0...𝐾))
9	3, 8	fssresd 6637	. . . 4 ⊢ (𝜑 → (𝐹 ↾ {0}):{0}⟶𝑉)
10		eqidd 2740	. . . . 5 ⊢ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0)
11		0nn0 12231	. . . . . 6 ⊢ 0 ∈ ℕ0
12		fveqeq2 6777	. . . . . . . 8 ⊢ (𝑥 = 0 → (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦)))
13		eqeq1 2743	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
14	12, 13	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = 0 → ((((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) → 0 = 𝑦)))
15		fveq2 6768	. . . . . . . . 9 ⊢ (𝑦 = 0 → ((𝐹 ↾ {0})‘𝑦) = ((𝐹 ↾ {0})‘0))
16	15	eqeq2d 2750	. . . . . . . 8 ⊢ (𝑦 = 0 → (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0)))
17		eqeq2 2751	. . . . . . . 8 ⊢ (𝑦 = 0 → (0 = 𝑦 ↔ 0 = 0))
18	16, 17	imbi12d 344	. . . . . . 7 ⊢ (𝑦 = 0 → ((((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) → 0 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0)))
19	14, 18	2ralsng 4617	. . . . . 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)))
20	11, 11, 19	mp2an 688	. . . . 5 ⊢ (∀𝑥 ∈ {0}∀𝑦 ∈ {0} (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0))
21	10, 20	mpbir 230	. . . 4 ⊢ ∀𝑥 ∈ {0}∀𝑦 ∈ {0} (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦)
22		dff13 7122	. . . 4 ⊢ ((𝐹 ↾ {0}):{0}–1-1→𝑉 ↔ ((𝐹 ↾ {0}):{0}⟶𝑉 ∧ ∀𝑥 ∈ {0}∀𝑦 ∈ {0} (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦)))
23	9, 21, 22	sylanblrc 589	. . 3 ⊢ (𝜑 → (𝐹 ↾ {0}):{0}–1-1→𝑉)
24		uncom 4091	. . . . . . . 8 ⊢ ((1...𝐾) ∪ {0}) = ({0} ∪ (1...𝐾))
25		fz0sn0fz1 13355	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ0 → (0...𝐾) = ({0} ∪ (1...𝐾)))
26	5, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0...𝐾) = ({0} ∪ (1...𝐾)))
27	24, 26	eqtr4id 2798	. . . . . . 7 ⊢ (𝜑 → ((1...𝐾) ∪ {0}) = (0...𝐾))
28		0nelfz1 13257	. . . . . . . . . 10 ⊢ 0 ∉ (1...𝐾)
29	28	neli 3052	. . . . . . . . 9 ⊢ ¬ 0 ∈ (1...𝐾)
30		disjsn 4652	. . . . . . . . 9 ⊢ (((1...𝐾) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...𝐾))
31	29, 30	mpbir 230	. . . . . . . 8 ⊢ ((1...𝐾) ∩ {0}) = ∅
32		uneqdifeq 4428	. . . . . . . 8 ⊢ (((1...𝐾) ⊆ (0...𝐾) ∧ ((1...𝐾) ∩ {0}) = ∅) → (((1...𝐾) ∪ {0}) = (0...𝐾) ↔ ((0...𝐾) ∖ (1...𝐾)) = {0}))
33	1, 31, 32	mp2an 688	. . . . . . 7 ⊢ (((1...𝐾) ∪ {0}) = (0...𝐾) ↔ ((0...𝐾) ∖ (1...𝐾)) = {0})
34	27, 33	sylib 217	. . . . . 6 ⊢ (𝜑 → ((0...𝐾) ∖ (1...𝐾)) = {0})
35	34	eqcomd 2745	. . . . 5 ⊢ (𝜑 → {0} = ((0...𝐾) ∖ (1...𝐾)))
36	35	reseq2d 5888	. . . 4 ⊢ (𝜑 → (𝐹 ↾ {0}) = (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))))
37		eqidd 2740	. . . 4 ⊢ (𝜑 → 𝑉 = 𝑉)
38	36, 35, 37	f1eq123d 6704	. . 3 ⊢ (𝜑 → ((𝐹 ↾ {0}):{0}–1-1→𝑉 ↔ (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))):((0...𝐾) ∖ (1...𝐾))–1-1→𝑉))
39	23, 38	mpbid 231	. 2 ⊢ (𝜑 → (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))):((0...𝐾) ∖ (1...𝐾))–1-1→𝑉)
40	35	imaeq2d 5966	. . . 4 ⊢ (𝜑 → (𝐹 “ {0}) = (𝐹 “ ((0...𝐾) ∖ (1...𝐾))))
41	40	ineq2d 4151	. . 3 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))))
42		incom 4139	. . . 4 ⊢ ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0}))
43		f1resfz0f1d.4	. . . 4 ⊢ (𝜑 → ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ∅)
44	42, 43	eqtr3id 2793	. . 3 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ∅)
45	41, 44	eqtr3d 2781	. 2 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))) = ∅)
46	2, 3, 4, 39, 45	f1resrcmplf1d 33038	1 ⊢ (𝜑 → 𝐹:(0...𝐾)–1-1→𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2109  ∀wral 3065   ∖ cdif 3888   ∪ cun 3889   ∩ cin 3890   ⊆ wss 3891  ∅c0 4261  {csn 4566   ↾ cres 5590   “ cima 5591  ⟶wf 6426  –1-1→wf1 6427  ‘cfv 6430  (class class class)co 7268  0cc0 10855  1c1 10856  ℕ₀cn0 12216  ...cfz 13221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-1st 7817  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-er 8472  df-en 8708  df-dom 8709  df-sdom 8710  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-nn 11957  df-n0 12217  df-z 12303  df-uz 12565  df-fz 13222

This theorem is referenced by:  pthhashvtx  33068