Theorem f1resfz0f1d 33704

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 13537	. . 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 13538	. . . . . 6 ⊢ (𝐾 ∈ ℕ0 → 0 ∈ (0...𝐾))
7		snssi 4768	. . . . . 6 ⊢ (0 ∈ (0...𝐾) → {0} ⊆ (0...𝐾))
8	5, 6, 7	3syl 18	. . . . 5 ⊢ (𝜑 → {0} ⊆ (0...𝐾))
9	3, 8	fssresd 6709	. . . 4 ⊢ (𝜑 → (𝐹 ↾ {0}):{0}⟶𝑉)
10		eqidd 2737	. . . . 5 ⊢ (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0) → 0 = 0)
11		0nn0 12428	. . . . . 6 ⊢ 0 ∈ ℕ0
12		fveqeq2 6851	. . . . . . . 8 ⊢ (𝑥 = 0 → (((𝐹 ↾ {0})‘𝑥) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦)))
13		eqeq1 2740	. . . . . . . 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 6842	. . . . . . . . 9 ⊢ (𝑦 = 0 → ((𝐹 ↾ {0})‘𝑦) = ((𝐹 ↾ {0})‘0))
16	15	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑦 = 0 → (((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘𝑦) ↔ ((𝐹 ↾ {0})‘0) = ((𝐹 ↾ {0})‘0)))
17		eqeq2 2748	. . . . . . . 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 4637	. . . . . 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 690	. . . . 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 7202	. . . 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 590	. . 3 ⊢ (𝜑 → (𝐹 ↾ {0}):{0}–1-1→𝑉)
24		uncom 4113	. . . . . . . 8 ⊢ ((1...𝐾) ∪ {0}) = ({0} ∪ (1...𝐾))
25		fz0sn0fz1 13558	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ0 → (0...𝐾) = ({0} ∪ (1...𝐾)))
26	5, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0...𝐾) = ({0} ∪ (1...𝐾)))
27	24, 26	eqtr4id 2795	. . . . . . 7 ⊢ (𝜑 → ((1...𝐾) ∪ {0}) = (0...𝐾))
28		0nelfz1 13460	. . . . . . . . . 10 ⊢ 0 ∉ (1...𝐾)
29	28	neli 3051	. . . . . . . . 9 ⊢ ¬ 0 ∈ (1...𝐾)
30		disjsn 4672	. . . . . . . . 9 ⊢ (((1...𝐾) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...𝐾))
31	29, 30	mpbir 230	. . . . . . . 8 ⊢ ((1...𝐾) ∩ {0}) = ∅
32		uneqdifeq 4450	. . . . . . . 8 ⊢ (((1...𝐾) ⊆ (0...𝐾) ∧ ((1...𝐾) ∩ {0}) = ∅) → (((1...𝐾) ∪ {0}) = (0...𝐾) ↔ ((0...𝐾) ∖ (1...𝐾)) = {0}))
33	1, 31, 32	mp2an 690	. . . . . . 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 2742	. . . . 5 ⊢ (𝜑 → {0} = ((0...𝐾) ∖ (1...𝐾)))
36	35	reseq2d 5937	. . . 4 ⊢ (𝜑 → (𝐹 ↾ {0}) = (𝐹 ↾ ((0...𝐾) ∖ (1...𝐾))))
37		eqidd 2737	. . . 4 ⊢ (𝜑 → 𝑉 = 𝑉)
38	36, 35, 37	f1eq123d 6776	. . 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 6013	. . . 4 ⊢ (𝜑 → (𝐹 “ {0}) = (𝐹 “ ((0...𝐾) ∖ (1...𝐾))))
41	40	ineq2d 4172	. . 3 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))))
42		incom 4161	. . . 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 2790	. . 3 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ {0})) = ∅)
45	41, 44	eqtr3d 2778	. 2 ⊢ (𝜑 → ((𝐹 “ (1...𝐾)) ∩ (𝐹 “ ((0...𝐾) ∖ (1...𝐾)))) = ∅)
46	2, 3, 4, 39, 45	f1resrcmplf1d 33691	1 ⊢ (𝜑 → 𝐹:(0...𝐾)–1-1→𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  ∀wral 3064   ∖ cdif 3907   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  {csn 4586   ↾ cres 5635   “ cima 5636  ⟶wf 6492  –1-1→wf1 6493  ‘cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052  ℕ₀cn0 12413  ...cfz 13424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425

This theorem is referenced by:  pthhashvtx  33721