Theorem clwwlknon1 30026

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 7400	. . 3 ⊢ (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1)
3	2	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1))
4		clwwlknon 30019	. . 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 29970	. . . . 5 ⊢ (𝑤 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
7	6	anbi1i 624	. . . 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 2738	. . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = 𝑉
10	9	wrdeqi 14502	. . . . . . . . . 10 ⊢ Word (Vtx‘𝐺) = Word 𝑉
11	10	eleq2i 2820	. . . . . . . . 9 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
12	11	biimpi 216	. . . . . . . 8 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → 𝑤 ∈ Word 𝑉)
13	12	3ad2ant2 1134	. . . . . . 7 ⊢ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → 𝑤 ∈ Word 𝑉)
14	13	ad2antrl 728	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 ∈ Word 𝑉)
15	13	adantr 480	. . . . . . . . 9 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → 𝑤 ∈ Word 𝑉)
16		simpl1 1192	. . . . . . . . 9 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (♯‘𝑤) = 1)
17		simpr 484	. . . . . . . . 9 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋)
18	15, 16, 17	3jca 1128	. . . . . . . 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 481	. . . . . . 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 14579	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑤 = ⟨“𝑋”⟩ ↔ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋)))
21	20	adantr 480	. . . . . . 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 257	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 = ⟨“𝑋”⟩)
23		sneq 4599	. . . . . . . . . . . . 13 ⊢ ((𝑤‘0) = 𝑋 → {(𝑤‘0)} = {𝑋})
24		clwwlknon1.e	. . . . . . . . . . . . . . 15 ⊢ 𝐸 = (Edg‘𝐺)
25	24	eqcomi 2738	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = 𝐸
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑤‘0) = 𝑋 → (Edg‘𝐺) = 𝐸)
27	23, 26	eleq12d 2822	. . . . . . . . . . . 12 ⊢ ((𝑤‘0) = 𝑋 → ({(𝑤‘0)} ∈ (Edg‘𝐺) ↔ {𝑋} ∈ 𝐸))
28	27	biimpd 229	. . . . . . . . . . 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 1135	. . . . . . . 8 ⊢ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → ((𝑤‘0) = 𝑋 → (𝑋 ∈ 𝑉 → {𝑋} ∈ 𝐸)))
32	31	imp 406	. . . . . . 7 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑋 ∈ 𝑉 → {𝑋} ∈ 𝐸))
33	32	impcom 407	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → {𝑋} ∈ 𝐸)
34	14, 22, 33	jca32 515	. . . . 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 6858	. . . . . . . . . 10 ⊢ (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = (♯‘⟨“𝑋”⟩))
36		s1len 14571	. . . . . . . . . 10 ⊢ (♯‘⟨“𝑋”⟩) = 1
37	35, 36	eqtrdi 2780	. . . . . . . . 9 ⊢ (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = 1)
38	37	ad2antrl 728	. . . . . . . 8 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (♯‘𝑤) = 1)
39	38	adantl 481	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (♯‘𝑤) = 1)
40	8	wrdeqi 14502	. . . . . . . . . 10 ⊢ Word 𝑉 = Word (Vtx‘𝐺)
41	40	eleq2i 2820	. . . . . . . . 9 ⊢ (𝑤 ∈ Word 𝑉 ↔ 𝑤 ∈ Word (Vtx‘𝐺))
42	41	biimpi 216	. . . . . . . 8 ⊢ (𝑤 ∈ Word 𝑉 → 𝑤 ∈ Word (Vtx‘𝐺))
43	42	ad2antrl 728	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → 𝑤 ∈ Word (Vtx‘𝐺))
44		fveq1 6857	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ⟨“𝑋”⟩ → (𝑤‘0) = (⟨“𝑋”⟩‘0))
45		s1fv 14575	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ 𝑉 → (⟨“𝑋”⟩‘0) = 𝑋)
46	44, 45	sylan9eq 2784	. . . . . . . . . . . . . 14 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → (𝑤‘0) = 𝑋)
47	46	eqcomd 2735	. . . . . . . . . . . . 13 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → 𝑋 = (𝑤‘0))
48	47	sneqd 4601	. . . . . . . . . . . 12 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → {𝑋} = {(𝑤‘0)})
49	24	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → 𝐸 = (Edg‘𝐺))
50	48, 49	eleq12d 2822	. . . . . . . . . . 11 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → ({𝑋} ∈ 𝐸 ↔ {(𝑤‘0)} ∈ (Edg‘𝐺)))
51	50	biimpd 229	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → ({𝑋} ∈ 𝐸 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
52	51	impancom 451	. . . . . . . . 9 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸) → (𝑋 ∈ 𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
53	52	adantl 481	. . . . . . . 8 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋 ∈ 𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
54	53	impcom 407	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → {(𝑤‘0)} ∈ (Edg‘𝐺))
55	39, 43, 54	3jca 1128	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
56	46	ex 412	. . . . . . . 8 ⊢ (𝑤 = ⟨“𝑋”⟩ → (𝑋 ∈ 𝑉 → (𝑤‘0) = 𝑋))
57	56	ad2antrl 728	. . . . . . 7 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋 ∈ 𝑉 → (𝑤‘0) = 𝑋))
58	57	impcom 407	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (𝑤‘0) = 𝑋)
59	55, 58	jca 511	. . . . 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 800	. . . 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	bitrid 283	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (1 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
62	61	rabbidva2 3407	. 2 ⊢ (𝑋 ∈ 𝑉 → {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
63	3, 5, 62	3eqtrd 2768	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {crab 3405  {csn 4589  ‘cfv 6511  (class class class)co 7387  0cc0 11068  1c1 11069  ♯chash 14295  Word cword 14478  ⟨“cs1 14560  Vtxcvtx 28923  Edgcedg 28974   ClWWalksN cclwwlkn 29953  ClWWalksNOncclwwlknon 30016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-oadd 8438  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-n0 12443  df-xnn0 12516  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616  df-hash 14296  df-word 14479  df-lsw 14528  df-s1 14561  df-clwwlk 29911  df-clwwlkn 29954  df-clwwlknon 30017

This theorem is referenced by:  clwwlknon1loop  30027  clwwlknon1nloop  30028