Theorem clwwlknon1 30448

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 7423	. . 3 ⊢ (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1)
3	2	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1))
4		clwwlknon 30441	. . 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 30392	. . . . 5 ⊢ (𝑤 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
7	6	anbi1i 635	. . . 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 2772	. . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = 𝑉
10	9	wrdeqi 14570	. . . . . . . . . 10 ⊢ Word (Vtx‘𝐺) = Word 𝑉
11	10	eleq2i 2855	. . . . . . . . 9 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
12	11	biimpi 219	. . . . . . . 8 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → 𝑤 ∈ Word 𝑉)
13	12	3ad2ant2 1152	. . . . . . 7 ⊢ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → 𝑤 ∈ Word 𝑉)
14	13	ad2antrl 740	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 ∈ Word 𝑉)
15	13	adantr 485	. . . . . . . . 9 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → 𝑤 ∈ Word 𝑉)
16		simpl1 1210	. . . . . . . . 9 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (♯‘𝑤) = 1)
17		simpr 489	. . . . . . . . 9 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋)
18	15, 16, 17	3jca 1146	. . . . . . . 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 486	. . . . . . 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 14648	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑤 = ⟨“𝑋”⟩ ↔ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋)))
21	20	adantr 485	. . . . . . 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 260	. . . . . 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 2772	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = 𝐸
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑤‘0) = 𝑋 → (Edg‘𝐺) = 𝐸)
27	23, 26	eleq12d 2857	. . . . . . . . . . . 12 ⊢ ((𝑤‘0) = 𝑋 → ({(𝑤‘0)} ∈ (Edg‘𝐺) ↔ {𝑋} ∈ 𝐸))
28	27	biimpd 232	. . . . . . . . . . 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 89	. . . . . . . . 9 ⊢ ({(𝑤‘0)} ∈ (Edg‘𝐺) → ((𝑤‘0) = 𝑋 → (𝑋 ∈ 𝑉 → {𝑋} ∈ 𝐸)))
31	30	3ad2ant3 1153	. . . . . . . 8 ⊢ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → ((𝑤‘0) = 𝑋 → (𝑋 ∈ 𝑉 → {𝑋} ∈ 𝐸)))
32	31	imp 411	. . . . . . 7 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑋 ∈ 𝑉 → {𝑋} ∈ 𝐸))
33	32	impcom 412	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → {𝑋} ∈ 𝐸)
34	14, 22, 33	jca32 524	. . . . 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 6881	. . . . . . . . . 10 ⊢ (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = (♯‘⟨“𝑋”⟩))
36		s1len 14640	. . . . . . . . . 10 ⊢ (♯‘⟨“𝑋”⟩) = 1
37	35, 36	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = 1)
38	37	ad2antrl 740	. . . . . . . 8 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (♯‘𝑤) = 1)
39	38	adantl 486	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (♯‘𝑤) = 1)
40	8	wrdeqi 14570	. . . . . . . . . 10 ⊢ Word 𝑉 = Word (Vtx‘𝐺)
41	40	eleq2i 2855	. . . . . . . . 9 ⊢ (𝑤 ∈ Word 𝑉 ↔ 𝑤 ∈ Word (Vtx‘𝐺))
42	41	biimpi 219	. . . . . . . 8 ⊢ (𝑤 ∈ Word 𝑉 → 𝑤 ∈ Word (Vtx‘𝐺))
43	42	ad2antrl 740	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → 𝑤 ∈ Word (Vtx‘𝐺))
44		fveq1 6880	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ⟨“𝑋”⟩ → (𝑤‘0) = (⟨“𝑋”⟩‘0))
45		s1fv 14644	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ 𝑉 → (⟨“𝑋”⟩‘0) = 𝑋)
46	44, 45	sylan9eq 2818	. . . . . . . . . . . . . 14 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → (𝑤‘0) = 𝑋)
47	46	eqcomd 2769	. . . . . . . . . . . . 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 2857	. . . . . . . . . . 11 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → ({𝑋} ∈ 𝐸 ↔ {(𝑤‘0)} ∈ (Edg‘𝐺)))
51	50	biimpd 232	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → ({𝑋} ∈ 𝐸 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
52	51	impancom 456	. . . . . . . . 9 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸) → (𝑋 ∈ 𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
53	52	adantl 486	. . . . . . . 8 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋 ∈ 𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
54	53	impcom 412	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → {(𝑤‘0)} ∈ (Edg‘𝐺))
55	39, 43, 54	3jca 1146	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
56	46	ex 417	. . . . . . . 8 ⊢ (𝑤 = ⟨“𝑋”⟩ → (𝑋 ∈ 𝑉 → (𝑤‘0) = 𝑋))
57	56	ad2antrl 740	. . . . . . 7 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋 ∈ 𝑉 → (𝑤‘0) = 𝑋))
58	57	impcom 412	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (𝑤‘0) = 𝑋)
59	55, 58	jca 520	. . . . 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 812	. . . 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 286	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (1 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
62	61	rabbidva2 3418	. 2 ⊢ (𝑋 ∈ 𝑉 → {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
63	3, 5, 62	3eqtrd 2802	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  {crab 3416  {csn 4589  ‘cfv 6536  (class class class)co 7410  0cc0 11095  1c1 11096  ♯chash 14362  Word cword 14546  ⟨“cs1 14629  Vtxcvtx 29346  Edgcedg 29397   ClWWalksN cclwwlkn 30375  ClWWalksNOncclwwlknon 30438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-n0 12500  df-xnn0 12573  df-z 12587  df-uz 12858  df-fz 13531  df-fzo 13679  df-hash 14363  df-word 14547  df-lsw 14596  df-s1 14630  df-clwwlk 30333  df-clwwlkn 30376  df-clwwlknon 30439

This theorem is referenced by:  clwwlknon1loop  30449  clwwlknon1nloop  30450