Theorem clwwlkvbij 26329

Description: There is a bijection between the set of closed walks of a fixed length starting at a fixed vertex represented by walks (as word) and the set of closed walks (as words) of a fixed length starting at a fixed vertex. The difference between these two representations is that in the first case the starting 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.)

Assertion

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

Distinct variable groups:   𝑓,𝐸,𝑤   𝑓,𝑁,𝑤   𝑆,𝑓,𝑤   𝑓,𝑉,𝑤   𝑓,𝑋,𝑤   𝑓,𝑌,𝑤

Proof of Theorem clwwlkvbij

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

Step	Hyp	Ref	Expression
1		fvex 6113	. . . . . 6 ⊢ ((𝑉 WWalksN 𝐸)‘𝑁) ∈ V
2	1	mptrabex 6392	. . . . 5 ⊢ (𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ∈ V
3	2	resex 5363	. . . 4 ⊢ ((𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ↾ {𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}) ∈ V
4		eqid 2610	. . . . 5 ⊢ (𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) = (𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩))
5		eqid 2610	. . . . . 6 ⊢ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} = {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)}
6	5, 4	clwwlkf1o 26326	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)):{𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)}–1-1-onto→((𝑉 ClWWalksN 𝐸)‘𝑁))
7		fveq1 6102	. . . . . . . 8 ⊢ (𝑦 = (𝑤 substr ⟨0, 𝑁⟩) → (𝑦‘0) = ((𝑤 substr ⟨0, 𝑁⟩)‘0))
8	7	eqeq1d 2612	. . . . . . 7 ⊢ (𝑦 = (𝑤 substr ⟨0, 𝑁⟩) → ((𝑦‘0) = 𝑆 ↔ ((𝑤 substr ⟨0, 𝑁⟩)‘0) = 𝑆))
9	8	3ad2ant3 1077	. . . . . 6 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ 𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∧ 𝑦 = (𝑤 substr ⟨0, 𝑁⟩)) → ((𝑦‘0) = 𝑆 ↔ ((𝑤 substr ⟨0, 𝑁⟩)‘0) = 𝑆))
10		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → ( lastS ‘𝑥) = ( lastS ‘𝑤))
11		fveq1 6102	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝑥‘0) = (𝑤‘0))
12	10, 11	eqeq12d 2625	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (( lastS ‘𝑥) = (𝑥‘0) ↔ ( lastS ‘𝑤) = (𝑤‘0)))
13	12	elrab 3331	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↔ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑤) = (𝑤‘0)))
14		wwlknimp 26215	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸))
15		simpll 786	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑤 ∈ Word 𝑉)
16		nnz 11276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
17		uzid 11578	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁))
18	16, 17	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘𝑁))
19		peano2uz 11617	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ (ℤ≥‘𝑁) → (𝑁 + 1) ∈ (ℤ≥‘𝑁))
20	18, 19	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (ℤ≥‘𝑁))
21		elfz1end 12242	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
22	21	biimpi 205	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁))
23	20, 22	jca 553	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ (ℤ≥‘𝑁) ∧ 𝑁 ∈ (1...𝑁)))
24		fzss2 12252	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...(𝑁 + 1)))
25	24	sselda 3568	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 + 1) ∈ (ℤ≥‘𝑁) ∧ 𝑁 ∈ (1...𝑁)) → 𝑁 ∈ (1...(𝑁 + 1)))
26	23, 25	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
27	26	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1)))
28		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑤) = (𝑁 + 1) → (1...(#‘𝑤)) = (1...(𝑁 + 1)))
29	28	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑤) = (𝑁 + 1) → (𝑁 ∈ (1...(#‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
30	29	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 1)) → (𝑁 ∈ (1...(#‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
31	30	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ (1...(#‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
32	27, 31	mpbird 246	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(#‘𝑤)))
33	15, 32	jca 553	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑤 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑤))))
34	33	ex 449	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑤)))))
35	34	3adant3 1074	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) → (𝑁 ∈ ℕ → (𝑤 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑤)))))
36	14, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℕ → (𝑤 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑤)))))
37	36	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑤) = (𝑤‘0)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑤)))))
38	13, 37	sylbi 206	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} → (𝑁 ∈ ℕ → (𝑤 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑤)))))
39	38	com12 32	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} → (𝑤 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑤)))))
40	39	3ad2ant3 1077	. . . . . . . . . 10 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} → (𝑤 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑤)))))
41	40	imp 444	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ 𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)}) → (𝑤 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑤))))
42		swrd0fv0 13292	. . . . . . . . 9 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑤))) → ((𝑤 substr ⟨0, 𝑁⟩)‘0) = (𝑤‘0))
43	41, 42	syl 17	. . . . . . . 8 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ 𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)}) → ((𝑤 substr ⟨0, 𝑁⟩)‘0) = (𝑤‘0))
44	43	eqeq1d 2612	. . . . . . 7 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ 𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)}) → (((𝑤 substr ⟨0, 𝑁⟩)‘0) = 𝑆 ↔ (𝑤‘0) = 𝑆))
45	44	3adant3 1074	. . . . . 6 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ 𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∧ 𝑦 = (𝑤 substr ⟨0, 𝑁⟩)) → (((𝑤 substr ⟨0, 𝑁⟩)‘0) = 𝑆 ↔ (𝑤‘0) = 𝑆))
46	9, 45	bitrd 267	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ 𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∧ 𝑦 = (𝑤 substr ⟨0, 𝑁⟩)) → ((𝑦‘0) = 𝑆 ↔ (𝑤‘0) = 𝑆))
47	4, 6, 46	f1oresrab 6302	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ↾ {𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}):{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑦‘0) = 𝑆})
48		f1oeq1 6040	. . . . 5 ⊢ (𝑓 = ((𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ↾ {𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}) → (𝑓:{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑦‘0) = 𝑆} ↔ ((𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ↾ {𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}):{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑦‘0) = 𝑆}))
49	48	spcegv 3267	. . . 4 ⊢ (((𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ↾ {𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}) ∈ V → (((𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ↾ {𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}):{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑦‘0) = 𝑆} → ∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑦‘0) = 𝑆}))
50	3, 47, 49	mpsyl 66	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑦‘0) = 𝑆})
51		fveq1 6102	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
52	51	eqeq1d 2612	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑤‘0) = 𝑆 ↔ (𝑦‘0) = 𝑆))
53	52	cbvrabv 3172	. . . . 5 ⊢ {𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑆} = {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑦‘0) = 𝑆}
54		f1oeq3 6042	. . . . 5 ⊢ ({𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑆} = {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑦‘0) = 𝑆} → (𝑓:{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑆} ↔ 𝑓:{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑦‘0) = 𝑆}))
55	53, 54	mp1i 13	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (𝑓:{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑆} ↔ 𝑓:{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑦‘0) = 𝑆}))
56	55	exbidv 1837	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑆} ↔ ∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑦‘0) = 𝑆}))
57	50, 56	mpbird 246	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑆})
58		df-rab 2905	. . . . 5 ⊢ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)} = {𝑤 ∣ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆))}
59		anass 679	. . . . . . 7 ⊢ (((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑆) ↔ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)))
60	59	bicomi 213	. . . . . 6 ⊢ ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)) ↔ ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑆))
61	60	abbii 2726	. . . . 5 ⊢ {𝑤 ∣ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆))} = {𝑤 ∣ ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑆)}
62	13	bicomi 213	. . . . . . . 8 ⊢ ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑤) = (𝑤‘0)) ↔ 𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)})
63	62	anbi1i 727	. . . . . . 7 ⊢ (((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑆) ↔ (𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∧ (𝑤‘0) = 𝑆))
64	63	abbii 2726	. . . . . 6 ⊢ {𝑤 ∣ ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑆)} = {𝑤 ∣ (𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∧ (𝑤‘0) = 𝑆)}
65		df-rab 2905	. . . . . 6 ⊢ {𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆} = {𝑤 ∣ (𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∧ (𝑤‘0) = 𝑆)}
66	64, 65	eqtr4i 2635	. . . . 5 ⊢ {𝑤 ∣ ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑆)} = {𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}
67	58, 61, 66	3eqtri 2636	. . . 4 ⊢ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)} = {𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}
68		f1oeq2 6041	. . . 4 ⊢ ({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)} = {𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆} → (𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑆} ↔ 𝑓:{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑆}))
69	67, 68	mp1i 13	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑆} ↔ 𝑓:{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑆}))
70	69	exbidv 1837	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (∃𝑓 𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑆} ↔ ∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑆}))
71	57, 70	mpbird 246	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑆})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896  {crab 2900  Vcvv 3173  {cpr 4127  ⟨cop 4131   ↦ cmpt 4643  ran crn 5039   ↾ cres 5040  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕcn 10897  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   substr csubstr 13150   WWalksN cwwlkn 26206   ClWWalksN cclwwlkn 26277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-wwlk 26207  df-wwlkn 26208  df-clwwlk 26279  df-clwwlkn 26280

This theorem is referenced by:  numclwwlkqhash  26627