Theorem clwwlkvbij 27987

Description: There is a bijection between the set of closed walks of a fixed length 𝑁 on a fixed vertex 𝑋 represented by walks (as word) and the set of closed walks (as words) of the fixed length 𝑁 on the fixed vertex 𝑋. The difference between these two representations is that in the first case the fixed vertex is repeated at the end of the word, and in the second case it is not. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 7-Jul-2022.) (Proof shortened by AV, 2-Nov-2022.)

Assertion

Ref	Expression
clwwlkvbij	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))

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

Allowed substitution hint:   𝑉(𝑤)

Proof of Theorem clwwlkvbij

Dummy variables 𝑥 𝑦 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7181	. . . . 5 ⊢ (𝑁 WWalksN 𝐺) ∈ V
2	1	mptrabex 6977	. . . 4 ⊢ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) ∈ V
3	2	resex 5869	. . 3 ⊢ ((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}) ∈ V
4		eqid 2759	. . . . . 6 ⊢ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) = (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁))
5		eqid 2759	. . . . . . 7 ⊢ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} = {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)}
6	5, 4	clwwlkf1o 27925	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)):{𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)}–1-1-onto→(𝑁 ClWWalksN 𝐺))
7		fveq1 6655	. . . . . . . . 9 ⊢ (𝑦 = (𝑤 prefix 𝑁) → (𝑦‘0) = ((𝑤 prefix 𝑁)‘0))
8	7	eqeq1d 2761	. . . . . . . 8 ⊢ (𝑦 = (𝑤 prefix 𝑁) → ((𝑦‘0) = 𝑋 ↔ ((𝑤 prefix 𝑁)‘0) = 𝑋))
9	8	3ad2ant3 1133	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∧ 𝑦 = (𝑤 prefix 𝑁)) → ((𝑦‘0) = 𝑋 ↔ ((𝑤 prefix 𝑁)‘0) = 𝑋))
10		fveq2 6656	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (lastS‘𝑥) = (lastS‘𝑤))
11		fveq1 6655	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝑥‘0) = (𝑤‘0))
12	10, 11	eqeq12d 2775	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → ((lastS‘𝑥) = (𝑥‘0) ↔ (lastS‘𝑤) = (𝑤‘0)))
13	12	elrab 3603	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↔ (𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)))
14		eqid 2759	. . . . . . . . . . . . . . 15 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
15		eqid 2759	. . . . . . . . . . . . . . 15 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
16	14, 15	wwlknp 27718	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
17		simpll 767	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑤 ∈ Word (Vtx‘𝐺))
18		nnz 12033	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
19		uzid 12287	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁))
20		peano2uz 12331	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ≥‘𝑁) → (𝑁 + 1) ∈ (ℤ≥‘𝑁))
21	18, 19, 20	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (ℤ≥‘𝑁))
22		elfz1end 12976	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
23	22	biimpi 219	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁))
24		fzss2 12986	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...(𝑁 + 1)))
25	24	sselda 3893	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 + 1) ∈ (ℤ≥‘𝑁) ∧ 𝑁 ∈ (1...𝑁)) → 𝑁 ∈ (1...(𝑁 + 1)))
26	21, 23, 25	syl2anc 588	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
27	26	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1)))
28		oveq2 7156	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑤) = (𝑁 + 1) → (1...(♯‘𝑤)) = (1...(𝑁 + 1)))
29	28	eleq2d 2838	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑤) = (𝑁 + 1) → (𝑁 ∈ (1...(♯‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
30	29	adantl 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) → (𝑁 ∈ (1...(♯‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
31	30	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ (1...(♯‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
32	27, 31	mpbird 260	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(♯‘𝑤)))
33	17, 32	jca 516	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤))))
34	33	ex 417	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
35	34	3adant3 1130	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
36	16, 35	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
37	36	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
38	13, 37	sylbi 220	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
39	38	impcom 412	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)}) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤))))
40		pfxfv0 14091	. . . . . . . . . 10 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤))) → ((𝑤 prefix 𝑁)‘0) = (𝑤‘0))
41	39, 40	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)}) → ((𝑤 prefix 𝑁)‘0) = (𝑤‘0))
42	41	eqeq1d 2761	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)}) → (((𝑤 prefix 𝑁)‘0) = 𝑋 ↔ (𝑤‘0) = 𝑋))
43	42	3adant3 1130	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∧ 𝑦 = (𝑤 prefix 𝑁)) → (((𝑤 prefix 𝑁)‘0) = 𝑋 ↔ (𝑤‘0) = 𝑋))
44	9, 43	bitrd 282	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∧ 𝑦 = (𝑤 prefix 𝑁)) → ((𝑦‘0) = 𝑋 ↔ (𝑤‘0) = 𝑋))
45	4, 6, 44	f1oresrab 6878	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}):{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}–1-1-onto→{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑋})
46	45	adantl 486	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}):{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}–1-1-onto→{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑋})
47		clwwlknon 27964	. . . . . 6 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑋}
48	47	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑋})
49	48	f1oeq3d 6597	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}):{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ ((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}):{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}–1-1-onto→{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑋}))
50	46, 49	mpbird 260	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}):{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
51		f1oeq1 6588	. . . 4 ⊢ (𝑓 = ((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}) → (𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ ((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}):{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁)))
52	51	spcegv 3516	. . 3 ⊢ (((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}) ∈ V → (((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}):{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) → ∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁)))
53	3, 50, 52	mpsyl 68	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
54		df-rab 3080	. . . . 5 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} = {𝑤 ∣ (𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋))}
55		anass 473	. . . . . . 7 ⊢ (((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)))
56	55	bicomi 227	. . . . . 6 ⊢ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)) ↔ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑋))
57	56	abbii 2824	. . . . 5 ⊢ {𝑤 ∣ (𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋))} = {𝑤 ∣ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑋)}
58	13	bicomi 227	. . . . . . . 8 ⊢ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) ↔ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)})
59	58	anbi1i 627	. . . . . . 7 ⊢ (((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∧ (𝑤‘0) = 𝑋))
60	59	abbii 2824	. . . . . 6 ⊢ {𝑤 ∣ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑋)} = {𝑤 ∣ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∧ (𝑤‘0) = 𝑋)}
61		df-rab 3080	. . . . . 6 ⊢ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋} = {𝑤 ∣ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∧ (𝑤‘0) = 𝑋)}
62	60, 61	eqtr4i 2785	. . . . 5 ⊢ {𝑤 ∣ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑋)} = {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}
63	54, 57, 62	3eqtri 2786	. . . 4 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} = {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}
64		f1oeq2 6589	. . . 4 ⊢ ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} = {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋} → (𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁)))
65	63, 64	mp1i 13	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁)))
66	65	exbidv 1923	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ ∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑋}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁)))
67	53, 66	mpbird 260	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539  ∃wex 1782   ∈ wcel 2112  {cab 2736  ∀wral 3071  {crab 3075  Vcvv 3410  {cpr 4522   ↦ cmpt 5110   ↾ cres 5524  –1-1-onto→wf1o 6332  ‘cfv 6333  (class class class)co 7148  0cc0 10565  1c1 10566   + caddc 10568  ℕcn 11664  ℤcz 12010  ℤ_≥cuz 12272  ...cfz 12929  ..^cfzo 13072  ♯chash 13730  Word cword 13903  lastSclsw 13951   prefix cpfx 14069  Vtxcvtx 26878  Edgcedg 26929   WWalksN cwwlksn 27701   ClWWalksN cclwwlkn 27898  ClWWalksNOncclwwlknon 27961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457  ax-cnex 10621  ax-resscn 10622  ax-1cn 10623  ax-icn 10624  ax-addcl 10625  ax-addrcl 10626  ax-mulcl 10627  ax-mulrcl 10628  ax-mulcom 10629  ax-addass 10630  ax-mulass 10631  ax-distr 10632  ax-i2m1 10633  ax-1ne0 10634  ax-1rid 10635  ax-rnegex 10636  ax-rrecex 10637  ax-cnre 10638  ax-pre-lttri 10639  ax-pre-lttrn 10640  ax-pre-ltadd 10641  ax-pre-mulgt0 10642

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-tp 4525  df-op 4527  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7578  df-1st 7691  df-2nd 7692  df-wrecs 7955  df-recs 8016  df-rdg 8054  df-1o 8110  df-oadd 8114  df-er 8297  df-map 8416  df-en 8526  df-dom 8527  df-sdom 8528  df-fin 8529  df-card 9391  df-pnf 10705  df-mnf 10706  df-xr 10707  df-ltxr 10708  df-le 10709  df-sub 10900  df-neg 10901  df-nn 11665  df-n0 11925  df-xnn0 11997  df-z 12011  df-uz 12273  df-rp 12421  df-fz 12930  df-fzo 13073  df-hash 13731  df-word 13904  df-lsw 13952  df-concat 13960  df-s1 13987  df-substr 14040  df-pfx 14070  df-wwlks 27705  df-wwlksn 27706  df-clwwlk 27856  df-clwwlkn 27899  df-clwwlknon 27962

This theorem is referenced by:  numclwwlkqhash  28249