Theorem f1resfz0f1d 34103

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 13597	. . 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 13598	. . . . . 6 ⊢ (𝐾 ∈ ℕ0 → 0 ∈ (0...𝐾))
7		snssi 4812	. . . . . 6 ⊢ (0 ∈ (0...𝐾) → {0} ⊆ (0...𝐾))
8	5, 6, 7	3syl 18	. . . . 5 ⊢ (𝜑 → {0} ⊆ (0...𝐾))
9	3, 8	fssresd 6759	. . . 4 ⊢ (𝜑 → (𝐹 ↾ {0}):{0}⟶𝑉)
10		eqidd 2734	. . . . 5 ⊢ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0)
11		0nn0 12487	. . . . . 6 ⊢ 0 ∈ ℕ0
12		fveqeq2 6901	. . . . . . . 8 ⊢ (𝑥 = 0 → (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦)))
13		eqeq1 2737	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
14	12, 13	imbi12d 345	. . . . . . 7 ⊢ (𝑥 = 0 → ((((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) → 𝑥 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) → 0 = 𝑦)))
15		fveq2 6892	. . . . . . . . 9 ⊢ (𝑦 = 0 → ((𝐹 ↾ {0})‘𝑦) = ((𝐹 ↾ {0})‘0))
16	15	eqeq2d 2744	. . . . . . . 8 ⊢ (𝑦 = 0 → (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0)))
17		eqeq2 2745	. . . . . . . 8 ⊢ (𝑦 = 0 → (0 = 𝑦 ↔ 0 = 0))
18	16, 17	imbi12d 345	. . . . . . 7 ⊢ (𝑦 = 0 → ((((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) → 0 = 𝑦) ↔ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0)))
19	14, 18	2ralsng 4681	. . . . . 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 691	. . . . 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 7254	. . . 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 591	. . 3 ⊢ (𝜑 → (𝐹 ↾ {0}):{0}–1-1→𝑉)
24		uncom 4154	. . . . . . . 8 ⊢ ((1...𝐾) ∪ {0}) = ({0} ∪ (1...𝐾))
25		fz0sn0fz1 13618	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ0 → (0...𝐾) = ({0} ∪ (1...𝐾)))
26	5, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0...𝐾) = ({0} ∪ (1...𝐾)))
27	24, 26	eqtr4id 2792	. . . . . . 7 ⊢ (𝜑 → ((1...𝐾) ∪ {0}) = (0...𝐾))
28		0nelfz1 13520	. . . . . . . . . 10 ⊢ 0 ∉ (1...𝐾)
29	28	neli 3049	. . . . . . . . 9 ⊢ ¬ 0 ∈ (1...𝐾)
30		disjsn 4716	. . . . . . . . 9 ⊢ (((1...𝐾) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...𝐾))
31	29, 30	mpbir 230	. . . . . . . 8 ⊢ ((1...𝐾) ∩ {0}) = ∅
32		uneqdifeq 4493	. . . . . . . 8 ⊢ (((1...𝐾) ⊆ (0...𝐾) ∧ ((1...𝐾) ∩ {0}) = ∅) → (((1...𝐾) ∪ {0}) = (0...𝐾) ↔ ((0...𝐾) ∖ (1...𝐾)) = {0}))
33	1, 31, 32	mp2an 691	. . . . . . 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 2739	. . . . 5 ⊢ (𝜑 → {0} = ((0...𝐾) ∖ (1...𝐾)))
36	35	reseq2d 5982	. . . 4 ⊢ (𝜑 → (𝐹 ↾ {0}) = (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))))
37		eqidd 2734	. . . 4 ⊢ (𝜑 → 𝑉 = 𝑉)
38	36, 35, 37	f1eq123d 6826	. . 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 6060	. . . 4 ⊢ (𝜑 → (𝐹 “ {0}) = (𝐹 “ ((0...𝐾) ∖ (1...𝐾))))
41	40	ineq2d 4213	. . 3 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))))
42		incom 4202	. . . 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 2787	. . 3 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ∅)
45	41, 44	eqtr3d 2775	. 2 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))) = ∅)
46	2, 3, 4, 39, 45	f1resrcmplf1d 34090	1 ⊢ (𝜑 → 𝐹:(0...𝐾)–1-1→𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  ∀wral 3062   ∖ cdif 3946   ∪ cun 3947   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323  {csn 4629   ↾ cres 5679   “ cima 5680  ⟶wf 6540  –1-1→wf1 6541  ‘cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111  ℕ₀cn0 12472  ...cfz 13484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485

This theorem is referenced by:  pthhashvtx  34118