Theorem f1resfz0f1d 13840

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.) (Revised by Mingli Yuan, 15-Aug-2026.)

Hypotheses

Ref	Expression
f1resfz0f1d.1	⊢ (𝜑 → 𝐾 ∈ ℕ0)
f1resfz0f1d.2	⊢ (𝜑 → 𝐹:(0...𝐾)⟶𝑉)
f1resfz0f1d.3	⊢ (𝜑 → Fun ◡(𝐹 ↾ (1...𝐾)))
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 13670	. . 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	3, 2	fssresd 6749	. . 3 ⊢ (𝜑 → (𝐹 ↾ (1...𝐾)):(1...𝐾)⟶𝑉)
5		f1resfz0f1d.3	. . 3 ⊢ (𝜑 → Fun ◡(𝐹 ↾ (1...𝐾)))
6		df-f1 6545	. . . 4 ⊢ ((𝐹 ↾ (1...𝐾)):(1...𝐾)–1-1→𝑉 ↔ ((𝐹 ↾ (1...𝐾)):(1...𝐾)⟶𝑉 ∧ Fun ◡(𝐹 ↾ (1...𝐾))))
7	6	a1i 11	. . 3 ⊢ (𝜑 → ((𝐹 ↾ (1...𝐾)):(1...𝐾)–1-1→𝑉 ↔ ((𝐹 ↾ (1...𝐾)):(1...𝐾)⟶𝑉 ∧ Fun ◡(𝐹 ↾ (1...𝐾)))))
8	4, 5, 7	mpbir2and 726	. 2 ⊢ (𝜑 → (𝐹 ↾ (1...𝐾)):(1...𝐾)–1-1→𝑉)
9		f1resfz0f1d.1	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
10		0elfz 13671	. . . . . 6 ⊢ (𝐾 ∈ ℕ0 → 0 ∈ (0...𝐾))
11		snssi 4753	. . . . . 6 ⊢ (0 ∈ (0...𝐾) → {0} ⊆ (0...𝐾))
12	9, 10, 11	3syl 19	. . . . 5 ⊢ (𝜑 → {0} ⊆ (0...𝐾))
13	3, 12	fssresd 6749	. . . 4 ⊢ (𝜑 → (𝐹 ↾ {0}):{0}⟶𝑉)
14		eqidd 2766	. . . . 5 ⊢ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0)
15		0nn0 12536	. . . . . 6 ⊢ 0 ∈ ℕ0
16		fveqeq2 6894	. . . . . . . 8 ⊢ (𝑥 = 0 → (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦)))
17		eqeq1 2769	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
18	16, 17	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = 0 → ((((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) → 0 = 𝑦)))
19		fveq2 6885	. . . . . . . . 9 ⊢ (𝑦 = 0 → ((𝐹 ↾ {0})‘𝑦) = ((𝐹 ↾ {0})‘0))
20	19	eqeq2d 2776	. . . . . . . 8 ⊢ (𝑦 = 0 → (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0)))
21		eqeq2 2777	. . . . . . . 8 ⊢ (𝑦 = 0 → (0 = 𝑦 ↔ 0 = 0))
22	20, 21	imbi12d 347	. . . . . . 7 ⊢ (𝑦 = 0 → ((((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) → 0 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0)))
23	18, 22	2ralsng 4646	. . . . . 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)))
24	15, 15, 23	mp2an 705	. . . . 5 ⊢ (∀𝑥 ∈ {0}∀𝑦 ∈ {0} (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0))
25	14, 24	mpbir 234	. . . 4 ⊢ ∀𝑥 ∈ {0}∀𝑦 ∈ {0} (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦)
26		dff13 7257	. . . 4 ⊢ ((𝐹 ↾ {0}):{0}–1-1→𝑉 ↔ ((𝐹 ↾ {0}):{0}⟶𝑉 ∧ ∀𝑥 ∈ {0}∀𝑦 ∈ {0} (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦)))
27	13, 25, 26	sylanblrc 602	. . 3 ⊢ (𝜑 → (𝐹 ↾ {0}):{0}–1-1→𝑉)
28		uncom 4112	. . . . . . . 8 ⊢ ((1...𝐾) ∪ {0}) = ({0} ∪ (1...𝐾))
29		fz0sn0fz1 13692	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ0 → (0...𝐾) = ({0} ∪ (1...𝐾)))
30	9, 29	syl 18	. . . . . . . 8 ⊢ (𝜑 → (0...𝐾) = ({0} ∪ (1...𝐾)))
31	28, 30	eqtr4id 2819	. . . . . . 7 ⊢ (𝜑 → ((1...𝐾) ∪ {0}) = (0...𝐾))
32		0nelfz1 13589	. . . . . . . . . 10 ⊢ 0 ∉ (1...𝐾)
33	32	neli 3068	. . . . . . . . 9 ⊢ ¬ 0 ∈ (1...𝐾)
34		disjsn 4679	. . . . . . . . 9 ⊢ (((1...𝐾) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...𝐾))
35	33, 34	mpbir 234	. . . . . . . 8 ⊢ ((1...𝐾) ∩ {0}) = ∅
36		uneqdifeq 4455	. . . . . . . 8 ⊢ (((1...𝐾) ⊆ (0...𝐾) ∧ ((1...𝐾) ∩ {0}) = ∅) → (((1...𝐾) ∪ {0}) = (0...𝐾) ↔ ((0...𝐾) ∖ (1...𝐾)) = {0}))
37	1, 35, 36	mp2an 705	. . . . . . 7 ⊢ (((1...𝐾) ∪ {0}) = (0...𝐾) ↔ ((0...𝐾) ∖ (1...𝐾)) = {0})
38	31, 37	sylib 221	. . . . . 6 ⊢ (𝜑 → ((0...𝐾) ∖ (1...𝐾)) = {0})
39	38	eqcomd 2771	. . . . 5 ⊢ (𝜑 → {0} = ((0...𝐾) ∖ (1...𝐾)))
40	39	reseq2d 5980	. . . 4 ⊢ (𝜑 → (𝐹 ↾ {0}) = (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))))
41		eqidd 2766	. . . 4 ⊢ (𝜑 → 𝑉 = 𝑉)
42	40, 39, 41	f1eq123d 6816	. . 3 ⊢ (𝜑 → ((𝐹 ↾ {0}):{0}–1-1→𝑉 ↔ (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))):((0...𝐾) ∖ (1...𝐾))–1-1→𝑉))
43	27, 42	mpbid 235	. 2 ⊢ (𝜑 → (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))):((0...𝐾) ∖ (1...𝐾))–1-1→𝑉)
44	39	imaeq2d 6064	. . . 4 ⊢ (𝜑 → (𝐹 “ {0}) = (𝐹 “ ((0...𝐾) ∖ (1...𝐾))))
45	44	ineq2d 4173	. . 3 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))))
46		incom 4162	. . . 4 ⊢ ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0}))
47		f1resfz0f1d.4	. . . 4 ⊢ (𝜑 → ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ∅)
48	46, 47	eqtr3id 2814	. . 3 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ∅)
49	45, 48	eqtr3d 2802	. 2 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))) = ∅)
50	2, 3, 8, 43, 49	f1resrcmplf1d 7278	1 ⊢ (𝜑 → 𝐹:(0...𝐾)–1-1→𝑉)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  {csn 4591  ^◡ccnv 5662   ↾ cres 5665   “ cima 5666  Fun wfun 6534  ⟶wf 6536  –1-1→wf1 6537  ‘cfv 6540  (class class class)co 7419  0cc0 11117  1c1 11118  ℕ₀cn0 12521  ...cfz 13553

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-nn 12251  df-n0 12522  df-z 12609  df-uz 12881  df-fz 13554

This theorem is used by:  pthhashvtx  30144