Theorem clwwlknon1 27499

Description: The set of closed walks on vertex 𝑋 of length 1 in a graph 𝐺 as words over the set of vertices. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 24-Mar-2022.)

Hypotheses

Ref	Expression
clwwlknon1.v	⊢ 𝑉 = (Vtx‘𝐺)
clwwlknon1.c	⊢ 𝐶 = (ClWWalksNOn‘𝐺)
clwwlknon1.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
clwwlknon1	⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})

Distinct variable groups:   𝑤,𝐺   𝑤,𝑉   𝑤,𝑋

Allowed substitution hints:   𝐶(𝑤)   𝐸(𝑤)

Proof of Theorem clwwlknon1

Step	Hyp	Ref	Expression
1		clwwlknon1.c	. . . 4 ⊢ 𝐶 = (ClWWalksNOn‘𝐺)
2	1	oveqi 6935	. . 3 ⊢ (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1)
3	2	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1))
4		clwwlknon 27492	. . 3 ⊢ (𝑋(ClWWalksNOn‘𝐺)1) = {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}
5	4	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋(ClWWalksNOn‘𝐺)1) = {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
6		clwwlkn1 27431	. . . . 5 ⊢ (𝑤 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
7	6	anbi1i 617	. . . 4 ⊢ ((𝑤 ∈ (1 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋))
8		clwwlknon1.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Vtx‘𝐺)
9	8	eqcomi 2787	. . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = 𝑉
10	9	wrdeqi 13625	. . . . . . . . . 10 ⊢ Word (Vtx‘𝐺) = Word 𝑉
11	10	eleq2i 2851	. . . . . . . . 9 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
12	11	biimpi 208	. . . . . . . 8 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → 𝑤 ∈ Word 𝑉)
13	12	3ad2ant2 1125	. . . . . . 7 ⊢ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → 𝑤 ∈ Word 𝑉)
14	13	ad2antrl 718	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 ∈ Word 𝑉)
15	13	adantr 474	. . . . . . . . 9 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → 𝑤 ∈ Word 𝑉)
16		simpl1 1199	. . . . . . . . 9 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (♯‘𝑤) = 1)
17		simpr 479	. . . . . . . . 9 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋)
18	15, 16, 17	3jca 1119	. . . . . . . 8 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋))
19	18	adantl 475	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋))
20		wrdl1s1 13704	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑤 = ⟨“𝑋”⟩ ↔ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋)))
21	20	adantr 474	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → (𝑤 = ⟨“𝑋”⟩ ↔ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋)))
22	19, 21	mpbird 249	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 = ⟨“𝑋”⟩)
23		sneq 4408	. . . . . . . . . . . . 13 ⊢ ((𝑤‘0) = 𝑋 → {(𝑤‘0)} = {𝑋})
24		clwwlknon1.e	. . . . . . . . . . . . . . 15 ⊢ 𝐸 = (Edg‘𝐺)
25	24	eqcomi 2787	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = 𝐸
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑤‘0) = 𝑋 → (Edg‘𝐺) = 𝐸)
27	23, 26	eleq12d 2853	. . . . . . . . . . . 12 ⊢ ((𝑤‘0) = 𝑋 → ({(𝑤‘0)} ∈ (Edg‘𝐺) ↔ {𝑋} ∈ 𝐸))
28	27	biimpd 221	. . . . . . . . . . 11 ⊢ ((𝑤‘0) = 𝑋 → ({(𝑤‘0)} ∈ (Edg‘𝐺) → {𝑋} ∈ 𝐸))
29	28	a1i 11	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → ((𝑤‘0) = 𝑋 → ({(𝑤‘0)} ∈ (Edg‘𝐺) → {𝑋} ∈ 𝐸)))
30	29	com13 88	. . . . . . . . 9 ⊢ ({(𝑤‘0)} ∈ (Edg‘𝐺) → ((𝑤‘0) = 𝑋 → (𝑋 ∈ 𝑉 → {𝑋} ∈ 𝐸)))
31	30	3ad2ant3 1126	. . . . . . . 8 ⊢ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → ((𝑤‘0) = 𝑋 → (𝑋 ∈ 𝑉 → {𝑋} ∈ 𝐸)))
32	31	imp 397	. . . . . . 7 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑋 ∈ 𝑉 → {𝑋} ∈ 𝐸))
33	32	impcom 398	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → {𝑋} ∈ 𝐸)
34	14, 22, 33	jca32 511	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)))
35		fveq2 6446	. . . . . . . . . 10 ⊢ (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = (♯‘⟨“𝑋”⟩))
36		s1len 13696	. . . . . . . . . 10 ⊢ (♯‘⟨“𝑋”⟩) = 1
37	35, 36	syl6eq 2830	. . . . . . . . 9 ⊢ (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = 1)
38	37	ad2antrl 718	. . . . . . . 8 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (♯‘𝑤) = 1)
39	38	adantl 475	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (♯‘𝑤) = 1)
40	8	wrdeqi 13625	. . . . . . . . . 10 ⊢ Word 𝑉 = Word (Vtx‘𝐺)
41	40	eleq2i 2851	. . . . . . . . 9 ⊢ (𝑤 ∈ Word 𝑉 ↔ 𝑤 ∈ Word (Vtx‘𝐺))
42	41	biimpi 208	. . . . . . . 8 ⊢ (𝑤 ∈ Word 𝑉 → 𝑤 ∈ Word (Vtx‘𝐺))
43	42	ad2antrl 718	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → 𝑤 ∈ Word (Vtx‘𝐺))
44		fveq1 6445	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ⟨“𝑋”⟩ → (𝑤‘0) = (⟨“𝑋”⟩‘0))
45		s1fv 13700	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ 𝑉 → (⟨“𝑋”⟩‘0) = 𝑋)
46	44, 45	sylan9eq 2834	. . . . . . . . . . . . . 14 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → (𝑤‘0) = 𝑋)
47	46	eqcomd 2784	. . . . . . . . . . . . 13 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → 𝑋 = (𝑤‘0))
48	47	sneqd 4410	. . . . . . . . . . . 12 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → {𝑋} = {(𝑤‘0)})
49	24	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → 𝐸 = (Edg‘𝐺))
50	48, 49	eleq12d 2853	. . . . . . . . . . 11 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → ({𝑋} ∈ 𝐸 ↔ {(𝑤‘0)} ∈ (Edg‘𝐺)))
51	50	biimpd 221	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → ({𝑋} ∈ 𝐸 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
52	51	impancom 445	. . . . . . . . 9 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸) → (𝑋 ∈ 𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
53	52	adantl 475	. . . . . . . 8 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋 ∈ 𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
54	53	impcom 398	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → {(𝑤‘0)} ∈ (Edg‘𝐺))
55	39, 43, 54	3jca 1119	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
56	46	ex 403	. . . . . . . 8 ⊢ (𝑤 = ⟨“𝑋”⟩ → (𝑋 ∈ 𝑉 → (𝑤‘0) = 𝑋))
57	56	ad2antrl 718	. . . . . . 7 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋 ∈ 𝑉 → (𝑤‘0) = 𝑋))
58	57	impcom 398	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (𝑤‘0) = 𝑋)
59	55, 58	jca 507	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋))
60	34, 59	impbida 791	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
61	7, 60	syl5bb 275	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (1 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
62	61	rabbidva2 3383	. 2 ⊢ (𝑋 ∈ 𝑉 → {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
63	3, 5, 62	3eqtrd 2818	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  {crab 3094  {csn 4398  ‘cfv 6135  (class class class)co 6922  0cc0 10272  1c1 10273  ♯chash 13435  Word cword 13599  ⟨“cs1 13685  Vtxcvtx 26344  Edgcedg 26395   ClWWalksN cclwwlkn 27413  ClWWalksNOncclwwlknon 27489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-hash 13436  df-word 13600  df-lsw 13653  df-s1 13686  df-clwwlk 27362  df-clwwlkn 27414  df-clwwlknon 27490

This theorem is referenced by:  clwwlknon1loop  27500  clwwlknon1nloop  27501