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Theorem clwwlksvbij 26788
 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.) (Revised by AV, 26-Apr-2021.)
Assertion
Ref Expression
clwwlksvbij (𝑁 ∈ ℕ → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆})
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑓,𝐺,𝑤   𝑓,𝑁   𝑆,𝑓,𝑤

Proof of Theorem clwwlksvbij
Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6632 . . . . . 6 (𝑁 WWalksN 𝐺) ∈ V
21mptrabex 6442 . . . . 5 (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ∈ V
32resex 5402 . . . 4 ((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}) ∈ V
4 eqid 2621 . . . . 5 (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) = (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩))
5 eqid 2621 . . . . . 6 {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} = {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)}
65, 4clwwlksf1o 26785 . . . . 5 (𝑁 ∈ ℕ → (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)):{𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)}–1-1-onto→(𝑁 ClWWalksN 𝐺))
7 fveq1 6147 . . . . . . . 8 (𝑦 = (𝑤 substr ⟨0, 𝑁⟩) → (𝑦‘0) = ((𝑤 substr ⟨0, 𝑁⟩)‘0))
87eqeq1d 2623 . . . . . . 7 (𝑦 = (𝑤 substr ⟨0, 𝑁⟩) → ((𝑦‘0) = 𝑆 ↔ ((𝑤 substr ⟨0, 𝑁⟩)‘0) = 𝑆))
983ad2ant3 1082 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∧ 𝑦 = (𝑤 substr ⟨0, 𝑁⟩)) → ((𝑦‘0) = 𝑆 ↔ ((𝑤 substr ⟨0, 𝑁⟩)‘0) = 𝑆))
10 fveq2 6148 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ( lastS ‘𝑥) = ( lastS ‘𝑤))
11 fveq1 6147 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝑥‘0) = (𝑤‘0))
1210, 11eqeq12d 2636 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (( lastS ‘𝑥) = (𝑥‘0) ↔ ( lastS ‘𝑤) = (𝑤‘0)))
1312elrab 3346 . . . . . . . . . . 11 (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↔ (𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑤) = (𝑤‘0)))
14 eqid 2621 . . . . . . . . . . . . . 14 (Vtx‘𝐺) = (Vtx‘𝐺)
15 eqid 2621 . . . . . . . . . . . . . 14 (Edg‘𝐺) = (Edg‘𝐺)
1614, 15wwlknp 26603 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
17 simpll 789 . . . . . . . . . . . . . . . 16 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑤 ∈ Word (Vtx‘𝐺))
18 nnz 11343 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
19 uzid 11646 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
20 peano2uz 11685 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (ℤ𝑁) → (𝑁 + 1) ∈ (ℤ𝑁))
2118, 19, 203syl 18 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (ℤ𝑁))
22 elfz1end 12313 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
2322biimpi 206 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁))
24 fzss2 12323 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 + 1) ∈ (ℤ𝑁) → (1...𝑁) ⊆ (1...(𝑁 + 1)))
2524sselda 3583 . . . . . . . . . . . . . . . . . . 19 (((𝑁 + 1) ∈ (ℤ𝑁) ∧ 𝑁 ∈ (1...𝑁)) → 𝑁 ∈ (1...(𝑁 + 1)))
2621, 23, 25syl2anc 692 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
2726adantl 482 . . . . . . . . . . . . . . . . 17 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1)))
28 oveq2 6612 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑤) = (𝑁 + 1) → (1...(#‘𝑤)) = (1...(𝑁 + 1)))
2928eleq2d 2684 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑤) = (𝑁 + 1) → (𝑁 ∈ (1...(#‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3029adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = (𝑁 + 1)) → (𝑁 ∈ (1...(#‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3130adantr 481 . . . . . . . . . . . . . . . . 17 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ (1...(#‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3227, 31mpbird 247 . . . . . . . . . . . . . . . 16 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(#‘𝑤)))
3317, 32jca 554 . . . . . . . . . . . . . . 15 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(#‘𝑤))))
3433ex 450 . . . . . . . . . . . . . 14 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(#‘𝑤)))))
35343adant3 1079 . . . . . . . . . . . . 13 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(#‘𝑤)))))
3616, 35syl 17 . . . . . . . . . . . 12 (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(#‘𝑤)))))
3736adantr 481 . . . . . . . . . . 11 ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑤) = (𝑤‘0)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(#‘𝑤)))))
3813, 37sylbi 207 . . . . . . . . . 10 (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(#‘𝑤)))))
3938impcom 446 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)}) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(#‘𝑤))))
40 swrd0fv0 13378 . . . . . . . . 9 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(#‘𝑤))) → ((𝑤 substr ⟨0, 𝑁⟩)‘0) = (𝑤‘0))
4139, 40syl 17 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)}) → ((𝑤 substr ⟨0, 𝑁⟩)‘0) = (𝑤‘0))
4241eqeq1d 2623 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)}) → (((𝑤 substr ⟨0, 𝑁⟩)‘0) = 𝑆 ↔ (𝑤‘0) = 𝑆))
43423adant3 1079 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∧ 𝑦 = (𝑤 substr ⟨0, 𝑁⟩)) → (((𝑤 substr ⟨0, 𝑁⟩)‘0) = 𝑆 ↔ (𝑤‘0) = 𝑆))
449, 43bitrd 268 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∧ 𝑦 = (𝑤 substr ⟨0, 𝑁⟩)) → ((𝑦‘0) = 𝑆 ↔ (𝑤‘0) = 𝑆))
454, 6, 44f1oresrab 6350 . . . 4 (𝑁 ∈ ℕ → ((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}):{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑆})
46 f1oeq1 6084 . . . . 5 (𝑓 = ((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}) → (𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑆} ↔ ((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}):{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑆}))
4746spcegv 3280 . . . 4 (((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}) ∈ V → (((𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ↾ {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}):{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑆} → ∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑆}))
483, 45, 47mpsyl 68 . . 3 (𝑁 ∈ ℕ → ∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑆})
49 fveq1 6147 . . . . . . 7 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
5049eqeq1d 2623 . . . . . 6 (𝑤 = 𝑦 → ((𝑤‘0) = 𝑆 ↔ (𝑦‘0) = 𝑆))
5150cbvrabv 3185 . . . . 5 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆} = {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑆}
52 f1oeq3 6086 . . . . 5 ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆} = {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑆} → (𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆} ↔ 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑆}))
5351, 52mp1i 13 . . . 4 (𝑁 ∈ ℕ → (𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆} ↔ 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑆}))
5453exbidv 1847 . . 3 (𝑁 ∈ ℕ → (∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆} ↔ ∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑆}))
5548, 54mpbird 247 . 2 (𝑁 ∈ ℕ → ∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆})
56 df-rab 2916 . . . . 5 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)} = {𝑤 ∣ (𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆))}
57 anass 680 . . . . . . 7 (((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑆) ↔ (𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)))
5857bicomi 214 . . . . . 6 ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)) ↔ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑆))
5958abbii 2736 . . . . 5 {𝑤 ∣ (𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆))} = {𝑤 ∣ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑆)}
6013bicomi 214 . . . . . . . 8 ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑤) = (𝑤‘0)) ↔ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)})
6160anbi1i 730 . . . . . . 7 (((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑆) ↔ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∧ (𝑤‘0) = 𝑆))
6261abbii 2736 . . . . . 6 {𝑤 ∣ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑆)} = {𝑤 ∣ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∧ (𝑤‘0) = 𝑆)}
63 df-rab 2916 . . . . . 6 {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆} = {𝑤 ∣ (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∧ (𝑤‘0) = 𝑆)}
6462, 63eqtr4i 2646 . . . . 5 {𝑤 ∣ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑆)} = {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}
6556, 59, 643eqtri 2647 . . . 4 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)} = {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}
66 f1oeq2 6085 . . . 4 ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)} = {𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆} → (𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆} ↔ 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆}))
6765, 66mp1i 13 . . 3 (𝑁 ∈ ℕ → (𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆} ↔ 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆}))
6867exbidv 1847 . 2 (𝑁 ∈ ℕ → (∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆} ↔ ∃𝑓 𝑓:{𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑥) = (𝑥‘0)} ∣ (𝑤‘0) = 𝑆}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆}))
6955, 68mpbird 247 1 (𝑁 ∈ ℕ → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987  {cab 2607  ∀wral 2907  {crab 2911  Vcvv 3186  {cpr 4150  ⟨cop 4154   ↦ cmpt 4673   ↾ cres 5076  –1-1-onto→wf1o 5846  ‘cfv 5847  (class class class)co 6604  0cc0 9880  1c1 9881   + caddc 9883  ℕcn 10964  ℤcz 11321  ℤ≥cuz 11631  ...cfz 12268  ..^cfzo 12406  #chash 13057  Word cword 13230   lastS clsw 13231   substr csubstr 13234  Vtxcvtx 25774  Edgcedg 25839   WWalksN cwwlksn 26587   ClWWalksN cclwwlksn 26743 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-lsw 13239  df-concat 13240  df-s1 13241  df-substr 13242  df-wwlks 26591  df-wwlksn 26592  df-clwwlks 26744  df-clwwlksn 26745 This theorem is referenced by:  numclwwlkqhash  27088
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