Theorem clwwlkvbij 29057

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 7390	. . . . 5 ⊢ (𝑁 WWalksN 𝐺) ∈ V
2	1	mptrabex 7175	. . . 4 ⊢ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) ∈ V
3	2	resex 5985	. . 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 2736	. . . . . 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 2736	. . . . . . 7 ⊢ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} = {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)}
6	5, 4	clwwlkf1o 28995	. . . . . 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 6841	. . . . . . . . 9 ⊢ (𝑦 = (𝑤 prefix 𝑁) → (𝑦‘0) = ((𝑤 prefix 𝑁)‘0))
8	7	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑦 = (𝑤 prefix 𝑁) → ((𝑦‘0) = 𝑋 ↔ ((𝑤 prefix 𝑁)‘0) = 𝑋))
9	8	3ad2ant3 1135	. . . . . . 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 6842	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (lastS‘𝑥) = (lastS‘𝑤))
11		fveq1 6841	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝑥‘0) = (𝑤‘0))
12	10, 11	eqeq12d 2752	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → ((lastS‘𝑥) = (𝑥‘0) ↔ (lastS‘𝑤) = (𝑤‘0)))
13	12	elrab 3645	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↔ (𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)))
14		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
15		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
16	14, 15	wwlknp 28788	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
17		simpll 765	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑤 ∈ Word (Vtx‘𝐺))
18		nnz 12520	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
19		uzid 12778	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁))
20		peano2uz 12826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ≥‘𝑁) → (𝑁 + 1) ∈ (ℤ≥‘𝑁))
21	18, 19, 20	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (ℤ≥‘𝑁))
22		elfz1end 13471	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
23	22	biimpi 215	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁))
24		fzss2 13481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...(𝑁 + 1)))
25	24	sselda 3944	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 + 1) ∈ (ℤ≥‘𝑁) ∧ 𝑁 ∈ (1...𝑁)) → 𝑁 ∈ (1...(𝑁 + 1)))
26	21, 23, 25	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
27	26	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1)))
28		oveq2 7365	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑤) = (𝑁 + 1) → (1...(♯‘𝑤)) = (1...(𝑁 + 1)))
29	28	eleq2d 2823	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑤) = (𝑁 + 1) → (𝑁 ∈ (1...(♯‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
30	29	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) → (𝑁 ∈ (1...(♯‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
31	30	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ (1...(♯‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
32	27, 31	mpbird 256	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(♯‘𝑤)))
33	17, 32	jca 512	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤))))
34	33	ex 413	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
35	34	3adant3 1132	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
38	13, 37	sylbi 216	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
39	38	impcom 408	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)}) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤))))
40		pfxfv0 14580	. . . . . . . . . 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 2738	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)}) → (((𝑤 prefix 𝑁)‘0) = 𝑋 ↔ (𝑤‘0) = 𝑋))
43	42	3adant3 1132	. . . . . . 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 278	. . . . . 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 7073	. . . . 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 482	. . . 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 29034	. . . . . 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 6781	. . . 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 256	. . 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 6772	. . . 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 3556	. . 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 3408	. . . . 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 469	. . . . . . 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 223	. . . . . 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 2806	. . . . 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 223	. . . . . . . 8 ⊢ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) ↔ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)})
59	58	anbi1i 624	. . . . . . 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 2806	. . . . . 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 3408	. . . . . 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 2767	. . . . 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 2768	. . . 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 6773	. . . 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 1924	. 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 256	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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2713  ∀wral 3064  {crab 3407  Vcvv 3445  {cpr 4588   ↦ cmpt 5188   ↾ cres 5635  –1-1-onto→wf1o 6495  ‘cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052   + caddc 11054  ℕcn 12153  ℤcz 12499  ℤ_≥cuz 12763  ...cfz 13424  ..^cfzo 13567  ♯chash 14230  Word cword 14402  lastSclsw 14450   prefix cpfx 14558  Vtxcvtx 27947  Edgcedg 27998   WWalksN cwwlksn 28771   ClWWalksN cclwwlkn 28968  ClWWalksNOncclwwlknon 29031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-wwlks 28775  df-wwlksn 28776  df-clwwlk 28926  df-clwwlkn 28969  df-clwwlknon 29032

This theorem is referenced by:  numclwwlkqhash  29319