Theorem clwwlksfo 41225

Description: Lemma 4 for clwwlksbij 41227: F is an onto function. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.)

Hypotheses

Ref	Expression
clwwlksbij.d	⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
clwwlksbij.f	⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))

Assertion

Ref	Expression
clwwlksfo	⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–onto→(𝑁 ClWWalkSN 𝐺))

Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑡,𝐷   𝑡,𝐺,𝑤   𝑡,𝑁

Allowed substitution hints:   𝐷(𝑤)   𝐹(𝑤,𝑡)

Proof of Theorem clwwlksfo

Dummy variables 𝑖 𝑥 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwwlksbij.d	. . 3 ⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
2		clwwlksbij.f	. . 3 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))
3	1, 2	clwwlksf 41222	. 2 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalkSN 𝐺))
4		eqid 2610	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2610	. . . . . . . 8 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
6	4, 5	clwwlknp 41195	. . . . . . 7 ⊢ (𝑝 ∈ (𝑁 ClWWalkSN 𝐺) → ((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)))
7		simpr 476	. . . . . . . . . 10 ⊢ ((((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
8		simpl1 1057	. . . . . . . . . 10 ⊢ ((((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)) ∧ 𝑁 ∈ ℕ) → (𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁))
9		3simpc 1053	. . . . . . . . . . 11 ⊢ (((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)))
10	9	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)) ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)))
11	1	clwwlksel 41221	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺))) → (𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷)
12	7, 8, 10, 11	syl3anc 1318	. . . . . . . . 9 ⊢ ((((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)) ∧ 𝑁 ∈ ℕ) → (𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷)
13		opeq2 4341	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = (#‘𝑝) → ⟨0, 𝑁⟩ = ⟨0, (#‘𝑝)⟩)
14	13	eqcoms 2618	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑝) = 𝑁 → ⟨0, 𝑁⟩ = ⟨0, (#‘𝑝)⟩)
15	14	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ ((#‘𝑝) = 𝑁 → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩) = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, (#‘𝑝)⟩))
16	15	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩) = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, (#‘𝑝)⟩))
17	16	3ad2ant1 1075	. . . . . . . . . . 11 ⊢ (((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩) = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, (#‘𝑝)⟩))
18	17	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)) ∧ 𝑁 ∈ ℕ) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩) = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, (#‘𝑝)⟩))
19		simpll 786	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑝 ∈ Word (Vtx‘𝐺))
20		fstwrdne0 13200	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁)) → (𝑝‘0) ∈ (Vtx‘𝐺))
21	20	ancoms 468	. . . . . . . . . . . . . 14 ⊢ (((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ 𝑁 ∈ ℕ) → (𝑝‘0) ∈ (Vtx‘𝐺))
22	21	s1cld 13236	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ 𝑁 ∈ ℕ) → ⟨“(𝑝‘0)”⟩ ∈ Word (Vtx‘𝐺))
23	19, 22	jca 553	. . . . . . . . . . . 12 ⊢ (((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ 𝑁 ∈ ℕ) → (𝑝 ∈ Word (Vtx‘𝐺) ∧ ⟨“(𝑝‘0)”⟩ ∈ Word (Vtx‘𝐺)))
24	23	3ad2antl1 1216	. . . . . . . . . . 11 ⊢ ((((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)) ∧ 𝑁 ∈ ℕ) → (𝑝 ∈ Word (Vtx‘𝐺) ∧ ⟨“(𝑝‘0)”⟩ ∈ Word (Vtx‘𝐺)))
25		swrdccat1 13309	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ Word (Vtx‘𝐺) ∧ ⟨“(𝑝‘0)”⟩ ∈ Word (Vtx‘𝐺)) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, (#‘𝑝)⟩) = 𝑝)
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ ((((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)) ∧ 𝑁 ∈ ℕ) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, (#‘𝑝)⟩) = 𝑝)
27	18, 26	eqtr2d 2645	. . . . . . . . 9 ⊢ ((((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)) ∧ 𝑁 ∈ ℕ) → 𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩))
28	12, 27	jca 553	. . . . . . . 8 ⊢ ((((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)) ∧ 𝑁 ∈ ℕ) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷 ∧ 𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩)))
29	28	ex 449	. . . . . . 7 ⊢ (((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝‘𝑖), (𝑝‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ (Edg‘𝐺)) → (𝑁 ∈ ℕ → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷 ∧ 𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩))))
30	6, 29	syl 17	. . . . . 6 ⊢ (𝑝 ∈ (𝑁 ClWWalkSN 𝐺) → (𝑁 ∈ ℕ → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷 ∧ 𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩))))
31	30	impcom 445	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ (𝑁 ClWWalkSN 𝐺)) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷 ∧ 𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩)))
32		oveq1 6556	. . . . . . 7 ⊢ (𝑥 = (𝑝 ++ ⟨“(𝑝‘0)”⟩) → (𝑥 substr ⟨0, 𝑁⟩) = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩))
33	32	eqeq2d 2620	. . . . . 6 ⊢ (𝑥 = (𝑝 ++ ⟨“(𝑝‘0)”⟩) → (𝑝 = (𝑥 substr ⟨0, 𝑁⟩) ↔ 𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩)))
34	33	rspcev 3282	. . . . 5 ⊢ (((𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷 ∧ 𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩)) → ∃𝑥 ∈ 𝐷 𝑝 = (𝑥 substr ⟨0, 𝑁⟩))
35	31, 34	syl 17	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ (𝑁 ClWWalkSN 𝐺)) → ∃𝑥 ∈ 𝐷 𝑝 = (𝑥 substr ⟨0, 𝑁⟩))
36	1, 2	clwwlksfv 41223	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (𝐹‘𝑥) = (𝑥 substr ⟨0, 𝑁⟩))
37	36	eqeq2d 2620	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (𝑝 = (𝐹‘𝑥) ↔ 𝑝 = (𝑥 substr ⟨0, 𝑁⟩)))
38	37	adantl 481	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑝 ∈ (𝑁 ClWWalkSN 𝐺)) ∧ 𝑥 ∈ 𝐷) → (𝑝 = (𝐹‘𝑥) ↔ 𝑝 = (𝑥 substr ⟨0, 𝑁⟩)))
39	38	rexbidva 3031	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ (𝑁 ClWWalkSN 𝐺)) → (∃𝑥 ∈ 𝐷 𝑝 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐷 𝑝 = (𝑥 substr ⟨0, 𝑁⟩)))
40	35, 39	mpbird 246	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ (𝑁 ClWWalkSN 𝐺)) → ∃𝑥 ∈ 𝐷 𝑝 = (𝐹‘𝑥))
41	40	ralrimiva 2949	. 2 ⊢ (𝑁 ∈ ℕ → ∀𝑝 ∈ (𝑁 ClWWalkSN 𝐺)∃𝑥 ∈ 𝐷 𝑝 = (𝐹‘𝑥))
42		dffo3 6282	. 2 ⊢ (𝐹:𝐷–onto→(𝑁 ClWWalkSN 𝐺) ↔ (𝐹:𝐷⟶(𝑁 ClWWalkSN 𝐺) ∧ ∀𝑝 ∈ (𝑁 ClWWalkSN 𝐺)∃𝑥 ∈ 𝐷 𝑝 = (𝐹‘𝑥)))
43	3, 41, 42	sylanbrc 695	1 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–onto→(𝑁 ClWWalkSN 𝐺))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  {cpr 4127  ⟨cop 4131   ↦ cmpt 4643  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕcn 10897  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   ++ cconcat 13148  ⟨“cs1 13149   substr csubstr 13150  Vtxcvtx 25673  Edgcedga 25792   WWalkSN cwwlksn 41029   ClWWalkSN cclwwlksn 41184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-wwlks 41033  df-wwlksn 41034  df-clwwlks 41185  df-clwwlksn 41186

This theorem is referenced by:  clwwlksf1o  41226