Theorem clwwlknon1 29979

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 7432	. . 3 ⊢ (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1)
3	2	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1))
4		clwwlknon 29972	. . 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 29923	. . . . 5 ⊢ (𝑤 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
7	6	anbi1i 622	. . . 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 2734	. . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = 𝑉
10	9	wrdeqi 14523	. . . . . . . . . 10 ⊢ Word (Vtx‘𝐺) = Word 𝑉
11	10	eleq2i 2817	. . . . . . . . 9 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
12	11	biimpi 215	. . . . . . . 8 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → 𝑤 ∈ Word 𝑉)
13	12	3ad2ant2 1131	. . . . . . 7 ⊢ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → 𝑤 ∈ Word 𝑉)
14	13	ad2antrl 726	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 ∈ Word 𝑉)
15	13	adantr 479	. . . . . . . . 9 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → 𝑤 ∈ Word 𝑉)
16		simpl1 1188	. . . . . . . . 9 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (♯‘𝑤) = 1)
17		simpr 483	. . . . . . . . 9 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋)
18	15, 16, 17	3jca 1125	. . . . . . . 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 480	. . . . . . 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 14600	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑤 = ⟨“𝑋”⟩ ↔ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋)))
21	20	adantr 479	. . . . . . 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 256	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 = ⟨“𝑋”⟩)
23		sneq 4640	. . . . . . . . . . . . 13 ⊢ ((𝑤‘0) = 𝑋 → {(𝑤‘0)} = {𝑋})
24		clwwlknon1.e	. . . . . . . . . . . . . . 15 ⊢ 𝐸 = (Edg‘𝐺)
25	24	eqcomi 2734	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = 𝐸
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑤‘0) = 𝑋 → (Edg‘𝐺) = 𝐸)
27	23, 26	eleq12d 2819	. . . . . . . . . . . 12 ⊢ ((𝑤‘0) = 𝑋 → ({(𝑤‘0)} ∈ (Edg‘𝐺) ↔ {𝑋} ∈ 𝐸))
28	27	biimpd 228	. . . . . . . . . . 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 1132	. . . . . . . 8 ⊢ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → ((𝑤‘0) = 𝑋 → (𝑋 ∈ 𝑉 → {𝑋} ∈ 𝐸)))
32	31	imp 405	. . . . . . 7 ⊢ ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑋 ∈ 𝑉 → {𝑋} ∈ 𝐸))
33	32	impcom 406	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → {𝑋} ∈ 𝐸)
34	14, 22, 33	jca32 514	. . . . 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 6896	. . . . . . . . . 10 ⊢ (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = (♯‘⟨“𝑋”⟩))
36		s1len 14592	. . . . . . . . . 10 ⊢ (♯‘⟨“𝑋”⟩) = 1
37	35, 36	eqtrdi 2781	. . . . . . . . 9 ⊢ (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = 1)
38	37	ad2antrl 726	. . . . . . . 8 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (♯‘𝑤) = 1)
39	38	adantl 480	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (♯‘𝑤) = 1)
40	8	wrdeqi 14523	. . . . . . . . . 10 ⊢ Word 𝑉 = Word (Vtx‘𝐺)
41	40	eleq2i 2817	. . . . . . . . 9 ⊢ (𝑤 ∈ Word 𝑉 ↔ 𝑤 ∈ Word (Vtx‘𝐺))
42	41	biimpi 215	. . . . . . . 8 ⊢ (𝑤 ∈ Word 𝑉 → 𝑤 ∈ Word (Vtx‘𝐺))
43	42	ad2antrl 726	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → 𝑤 ∈ Word (Vtx‘𝐺))
44		fveq1 6895	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ⟨“𝑋”⟩ → (𝑤‘0) = (⟨“𝑋”⟩‘0))
45		s1fv 14596	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ 𝑉 → (⟨“𝑋”⟩‘0) = 𝑋)
46	44, 45	sylan9eq 2785	. . . . . . . . . . . . . 14 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → (𝑤‘0) = 𝑋)
47	46	eqcomd 2731	. . . . . . . . . . . . 13 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → 𝑋 = (𝑤‘0))
48	47	sneqd 4642	. . . . . . . . . . . 12 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → {𝑋} = {(𝑤‘0)})
49	24	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → 𝐸 = (Edg‘𝐺))
50	48, 49	eleq12d 2819	. . . . . . . . . . 11 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → ({𝑋} ∈ 𝐸 ↔ {(𝑤‘0)} ∈ (Edg‘𝐺)))
51	50	biimpd 228	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → ({𝑋} ∈ 𝐸 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
52	51	impancom 450	. . . . . . . . 9 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸) → (𝑋 ∈ 𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
53	52	adantl 480	. . . . . . . 8 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋 ∈ 𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
54	53	impcom 406	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → {(𝑤‘0)} ∈ (Edg‘𝐺))
55	39, 43, 54	3jca 1125	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
56	46	ex 411	. . . . . . . 8 ⊢ (𝑤 = ⟨“𝑋”⟩ → (𝑋 ∈ 𝑉 → (𝑤‘0) = 𝑋))
57	56	ad2antrl 726	. . . . . . 7 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋 ∈ 𝑉 → (𝑤‘0) = 𝑋))
58	57	impcom 406	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (𝑤‘0) = 𝑋)
59	55, 58	jca 510	. . . . 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 799	. . . 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 282	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (1 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
62	61	rabbidva2 3420	. 2 ⊢ (𝑋 ∈ 𝑉 → {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
63	3, 5, 62	3eqtrd 2769	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3418  {csn 4630  ‘cfv 6549  (class class class)co 7419  0cc0 11140  1c1 11141  ♯chash 14325  Word cword 14500  ⟨“cs1 14581  Vtxcvtx 28881  Edgcedg 28932   ClWWalksN cclwwlkn 29906  ClWWalksNOncclwwlknon 29969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-n0 12506  df-xnn0 12578  df-z 12592  df-uz 12856  df-fz 13520  df-fzo 13663  df-hash 14326  df-word 14501  df-lsw 14549  df-s1 14582  df-clwwlk 29864  df-clwwlkn 29907  df-clwwlknon 29970

This theorem is referenced by:  clwwlknon1loop  29980  clwwlknon1nloop  29981