Theorem clwwlkvbij 30183

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 7400	. . . . 5 ⊢ (𝑁 WWalksN 𝐺) ∈ V
2	1	mptrabex 7180	. . . 4 ⊢ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) ∈ V
3	2	resex 5994	. . 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 30121	. . . . . 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 6839	. . . . . . . . 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 1136	. . . . . . 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 6840	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (lastS‘𝑥) = (lastS‘𝑤))
11		fveq1 6839	. . . . . . . . . . . . . 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 3634	. . . . . . . . . . . 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 29911	. . . . . . . . . . . . . 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 12545	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
19		uzid 12803	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁))
20		peano2uz 12851	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ≥‘𝑁) → (𝑁 + 1) ∈ (ℤ≥‘𝑁))
21	18, 19, 20	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (ℤ≥‘𝑁))
22		elfz1end 13508	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
23	22	biimpi 216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁))
24		fzss2 13518	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...(𝑁 + 1)))
25	24	sselda 3921	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 + 1) ∈ (ℤ≥‘𝑁) ∧ 𝑁 ∈ (1...𝑁)) → 𝑁 ∈ (1...(𝑁 + 1)))
26	21, 23, 25	syl2anc 585	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
27	26	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1)))
28		oveq2 7375	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑤) = (𝑁 + 1) → (1...(♯‘𝑤)) = (1...(𝑁 + 1)))
29	28	eleq2d 2822	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑤) = (𝑁 + 1) → (𝑁 ∈ (1...(♯‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
30	29	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) → (𝑁 ∈ (1...(♯‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
31	30	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ (1...(♯‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
32	27, 31	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(♯‘𝑤)))
33	17, 32	jca 511	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤))))
34	33	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
35	34	3adant3 1133	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
38	13, 37	sylbi 217	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
39	38	impcom 407	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)}) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤))))
40		pfxfv0 14654	. . . . . . . . . 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 1133	. . . . . . 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 279	. . . . . 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 7080	. . . . 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 481	. . . 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 30160	. . . . . 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 6777	. . . 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 257	. . 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 6768	. . . 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 3539	. . 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 3390	. . . . 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 468	. . . . . . 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 224	. . . . . 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 2803	. . . . 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 224	. . . . . . . 8 ⊢ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) ↔ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)})
59	58	anbi1i 625	. . . . . . 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 2803	. . . . . 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 3390	. . . . . 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 2762	. . . . 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 2763	. . . 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 6769	. . . 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 257	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2714  ∀wral 3051  {crab 3389  Vcvv 3429  {cpr 4569   ↦ cmpt 5166   ↾ cres 5633  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367  0cc0 11038  1c1 11039   + caddc 11041  ℕcn 12174  ℤcz 12524  ℤ_≥cuz 12788  ...cfz 13461  ..^cfzo 13608  ♯chash 14292  Word cword 14475  lastSclsw 14524   prefix cpfx 14633  Vtxcvtx 29065  Edgcedg 29116   WWalksN cwwlksn 29894   ClWWalksN cclwwlkn 30094  ClWWalksNOncclwwlknon 30157

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-hash 14293  df-word 14476  df-lsw 14525  df-concat 14533  df-s1 14559  df-substr 14604  df-pfx 14634  df-wwlks 29898  df-wwlksn 29899  df-clwwlk 30052  df-clwwlkn 30095  df-clwwlknon 30158

This theorem is referenced by:  numclwwlkqhash  30445