Theorem clwwlksfo 26918

Description: Lemma 4 for clwwlksbij 26920: 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 26915	. 2 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺))
4		eqid 2622	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2622	. . . . . . . 8 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
6	4, 5	clwwlknp 26887	. . . . . . 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 477	. . . . . . . . . 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 1064	. . . . . . . . . 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 1060	. . . . . . . . . . 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 481	. . . . . . . . . 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 26914	. . . . . . . . . 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 1326	. . . . . . . . 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 4403	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = (#‘𝑝) → ⟨0, 𝑁⟩ = ⟨0, (#‘𝑝)⟩)
14	13	eqcoms 2630	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑝) = 𝑁 → ⟨0, 𝑁⟩ = ⟨0, (#‘𝑝)⟩)
15	14	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ ((#‘𝑝) = 𝑁 → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩) = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, (#‘𝑝)⟩))
16	15	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩) = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, (#‘𝑝)⟩))
17	16	3ad2ant1 1082	. . . . . . . . . . 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 481	. . . . . . . . . 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 790	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑝 ∈ Word (Vtx‘𝐺))
20		fstwrdne0 13345	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁)) → (𝑝‘0) ∈ (Vtx‘𝐺))
21	20	ancoms 469	. . . . . . . . . . . . . 14 ⊢ (((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ 𝑁 ∈ ℕ) → (𝑝‘0) ∈ (Vtx‘𝐺))
22	21	s1cld 13383	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ 𝑁 ∈ ℕ) → ⟨“(𝑝‘0)”⟩ ∈ Word (Vtx‘𝐺))
23	19, 22	jca 554	. . . . . . . . . . . 12 ⊢ (((𝑝 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑝) = 𝑁) ∧ 𝑁 ∈ ℕ) → (𝑝 ∈ Word (Vtx‘𝐺) ∧ ⟨“(𝑝‘0)”⟩ ∈ Word (Vtx‘𝐺)))
24	23	3ad2antl1 1223	. . . . . . . . . . 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 13457	. . . . . . . . . . 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 2657	. . . . . . . . 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 554	. . . . . . . 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 450	. . . . . . 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 446	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ (𝑁 ClWWalksN 𝐺)) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷 ∧ 𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩)))
32		oveq1 6657	. . . . . . 7 ⊢ (𝑥 = (𝑝 ++ ⟨“(𝑝‘0)”⟩) → (𝑥 substr ⟨0, 𝑁⟩) = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩))
33	32	eqeq2d 2632	. . . . . 6 ⊢ (𝑥 = (𝑝 ++ ⟨“(𝑝‘0)”⟩) → (𝑝 = (𝑥 substr ⟨0, 𝑁⟩) ↔ 𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩)))
34	33	rspcev 3309	. . . . 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 26916	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (𝐹‘𝑥) = (𝑥 substr ⟨0, 𝑁⟩))
37	36	eqeq2d 2632	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (𝑝 = (𝐹‘𝑥) ↔ 𝑝 = (𝑥 substr ⟨0, 𝑁⟩)))
38	37	adantl 482	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑝 ∈ (𝑁 ClWWalksN 𝐺)) ∧ 𝑥 ∈ 𝐷) → (𝑝 = (𝐹‘𝑥) ↔ 𝑝 = (𝑥 substr ⟨0, 𝑁⟩)))
39	38	rexbidva 3049	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ (𝑁 ClWWalksN 𝐺)) → (∃𝑥 ∈ 𝐷 𝑝 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐷 𝑝 = (𝑥 substr ⟨0, 𝑁⟩)))
40	35, 39	mpbird 247	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑥 ∈ 𝐷 𝑝 = (𝐹‘𝑥))
41	40	ralrimiva 2966	. 2 ⊢ (𝑁 ∈ ℕ → ∀𝑝 ∈ (𝑁 ClWWalksN 𝐺)∃𝑥 ∈ 𝐷 𝑝 = (𝐹‘𝑥))
42		dffo3 6374	. 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 698	1 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–onto→(𝑁 ClWWalksN 𝐺))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  {crab 2916  {cpr 4179  ⟨cop 4183   ↦ cmpt 4729  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888  (class class class)co 6650  0cc0 9936  1c1 9937   + caddc 9939   − cmin 10266  ℕcn 11020  ..^cfzo 12465  #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293  ⟨“cs1 13294   substr csubstr 13295  Vtxcvtx 25874  Edgcedg 25939   WWalksN cwwlksn 26718   ClWWalksN cclwwlksn 26876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-wwlks 26722  df-wwlksn 26723  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  clwwlksf1o  26919