Theorem f1resfz0f1d 13967

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)⟶1-1𝑉)

Proof of Theorem f1resfz0f1d

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

Step	Hyp	Ref	Expression
1		fz1ssfz0 13797	. . 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 6779	. . 3 ⊢ (𝜑 → (𝐹 ↾ (1...𝐾)):(1...𝐾)⟶𝑉)
5		f1resfz0f1d.3	. . 3 ⊢ (𝜑 → Fun ◡(𝐹 ↾ (1...𝐾)))
6		df-f1 6571	. . . 4 ⊢ ((𝐹 ↾ (1...𝐾)):(1...𝐾)⟶1-1𝑉 ↔ ((𝐹 ↾ (1...𝐾)):(1...𝐾)⟶𝑉 ∧ Fun ◡(𝐹 ↾ (1...𝐾))))
7	6	a1i 11	. . 3 ⊢ (𝜑 → ((𝐹 ↾ (1...𝐾)):(1...𝐾)⟶1-1𝑉 ↔ ((𝐹 ↾ (1...𝐾)):(1...𝐾)⟶𝑉 ∧ Fun ◡(𝐹 ↾ (1...𝐾)))))
8	4, 5, 7	mpbir2and 726	. 2 ⊢ (𝜑 → (𝐹 ↾ (1...𝐾)):(1...𝐾)⟶1-1𝑉)
9		f1resfz0f1d.1	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
10		0elfz 13798	. . . . . 6 ⊢ (𝐾 ∈ ℕ0 → 0 ∈ (0...𝐾))
11		snssi 4748	. . . . . 6 ⊢ (0 ∈ (0...𝐾) → {0} ⊆ (0...𝐾))
12	9, 10, 11	3syl 19	. . . . 5 ⊢ (𝜑 → {0} ⊆ (0...𝐾))
13	3, 12	fssresd 6779	. . . 4 ⊢ (𝜑 → (𝐹 ↾ {0}):{0}⟶𝑉)
14		eqidd 2764	. . . . 5 ⊢ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0)
15		0nn0 12661	. . . . . 6 ⊢ 0 ∈ ℕ0
16		fveqeq2 6925	. . . . . . . 8 ⊢ (𝑥 = 0 → (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦)))
17		eqeq1 2767	. . . . . . . 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 6916	. . . . . . . . 9 ⊢ (𝑦 = 0 → ((𝐹 ↾ {0})‘𝑦) = ((𝐹 ↾ {0})‘0))
20	19	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑦 = 0 → (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0)))
21		eqeq2 2775	. . . . . . . 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 4640	. . . . . 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 7289	. . . 4 ⊢ ((𝐹 ↾ {0}):{0}⟶1-1𝑉 ↔ ((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}⟶1-1𝑉)
28		uncom 4160	. . . . . . . 8 ⊢ ((1...𝐾) ∪ {0}) = ({0} ∪ (1...𝐾))
29		fz0sn0fz1 13819	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ0 → (0...𝐾) = ({0} ∪ (1...𝐾)))
30	9, 29	syl 18	. . . . . . . 8 ⊢ (𝜑 → (0...𝐾) = ({0} ∪ (1...𝐾)))
31	28, 30	eqtr4id 2817	. . . . . . 7 ⊢ (𝜑 → ((1...𝐾) ∪ {0}) = (0...𝐾))
32		0nelfz1 13716	. . . . . . . . . 10 ⊢ 0 ∉ (1...𝐾)
33	32	neli 3067	. . . . . . . . 9 ⊢ ¬ 0 ∈ (1...𝐾)
34		disjsn 4673	. . . . . . . . 9 ⊢ (((1...𝐾) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...𝐾))
35	33, 34	mpbir 234	. . . . . . . 8 ⊢ ((1...𝐾) ∩ {0}) = ∅
36		uneqdifeq 4451	. . . . . . . 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 2769	. . . . 5 ⊢ (𝜑 → {0} = ((0...𝐾) ∖ (1...𝐾)))
40	39	reseq2d 5986	. . . 4 ⊢ (𝜑 → (𝐹 ↾ {0}) = (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))))
41		eqidd 2764	. . . 4 ⊢ (𝜑 → 𝑉 = 𝑉)
42	40, 39, 41	f1eq123d 6846	. . 3 ⊢ (𝜑 → ((𝐹 ↾ {0}):{0}⟶1-1𝑉 ↔ (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))):((0...𝐾) ∖ (1...𝐾))⟶1-1𝑉))
43	27, 42	mpbid 235	. 2 ⊢ (𝜑 → (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))):((0...𝐾) ∖ (1...𝐾))⟶1-1𝑉)
44	39	imaeq2d 6070	. . . 4 ⊢ (𝜑 → (𝐹 “ {0}) = (𝐹 “ ((0...𝐾) ∖ (1...𝐾))))
45	44	ineq2d 4120	. . 3 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))))
46		incom 4109	. . . 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 2812	. . 3 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ∅)
49	45, 48	eqtr3d 2800	. 2 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))) = ∅)
50	2, 3, 8, 43, 49	f1resrcmplf1d 7310	1 ⊢ (𝜑 → 𝐹:(0...𝐾)⟶1-1𝑉)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588  ^◡ccnv 5657   ↾ cres 5660   “ cima 5661  Fun wfun 6560  ⟶wf 6562  ⟶1-1wf1 6563  ‘cfv 6566  (class class class)co 7451  0cc0 11239  1c1 11240  ℕ₀cn0 12646  ...cfz 13679

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-n0 12647  df-z 12734  df-uz 13006  df-fz 13680

This theorem is used by:  pthhashvtx  30372