Theorem clwwlknon1 30126

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 7444	. . 3 ⊢ (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1)
3	2	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1))
4		clwwlknon 30119	. . 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 30070	. . . . 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 2744	. . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = 𝑉
10	9	wrdeqi 14572	. . . . . . . . . 10 ⊢ Word (Vtx‘𝐺) = Word 𝑉
11	10	eleq2i 2831	. . . . . . . . 9 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
12	11	biimpi 216	. . . . . . . 8 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → 𝑤 ∈ Word 𝑉)
13	12	3ad2ant2 1133	. . . . . . 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 1190	. . . . . . . . 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 1127	. . . . . . . 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 14649	. . . . . . . 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 4641	. . . . . . . . . . . . 13 ⊢ ((𝑤‘0) = 𝑋 → {(𝑤‘0)} = {𝑋})
24		clwwlknon1.e	. . . . . . . . . . . . . . 15 ⊢ 𝐸 = (Edg‘𝐺)
25	24	eqcomi 2744	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = 𝐸
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑤‘0) = 𝑋 → (Edg‘𝐺) = 𝐸)
27	23, 26	eleq12d 2833	. . . . . . . . . . . 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 1134	. . . . . . . 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 6907	. . . . . . . . . 10 ⊢ (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = (♯‘⟨“𝑋”⟩))
36		s1len 14641	. . . . . . . . . 10 ⊢ (♯‘⟨“𝑋”⟩) = 1
37	35, 36	eqtrdi 2791	. . . . . . . . 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 14572	. . . . . . . . . 10 ⊢ Word 𝑉 = Word (Vtx‘𝐺)
41	40	eleq2i 2831	. . . . . . . . 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 6906	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ⟨“𝑋”⟩ → (𝑤‘0) = (⟨“𝑋”⟩‘0))
45		s1fv 14645	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ 𝑉 → (⟨“𝑋”⟩‘0) = 𝑋)
46	44, 45	sylan9eq 2795	. . . . . . . . . . . . . 14 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → (𝑤‘0) = 𝑋)
47	46	eqcomd 2741	. . . . . . . . . . . . 13 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → 𝑋 = (𝑤‘0))
48	47	sneqd 4643	. . . . . . . . . . . 12 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → {𝑋} = {(𝑤‘0)})
49	24	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋 ∈ 𝑉) → 𝐸 = (Edg‘𝐺))
50	48, 49	eleq12d 2833	. . . . . . . . . . 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 1127	. . . . . 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 801	. . . 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 3435	. 2 ⊢ (𝑋 ∈ 𝑉 → {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
63	3, 5, 62	3eqtrd 2779	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {crab 3433  {csn 4631  ‘cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154  ♯chash 14366  Word cword 14549  ⟨“cs1 14630  Vtxcvtx 29028  Edgcedg 29079   ClWWalksN cclwwlkn 30053  ClWWalksNOncclwwlknon 30116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-lsw 14598  df-s1 14631  df-clwwlk 30011  df-clwwlkn 30054  df-clwwlknon 30117

This theorem is referenced by:  clwwlknon1loop  30127  clwwlknon1nloop  30128