MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwwlkvbij Structured version   Visualization version   GIF version

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 ((𝑋𝑉𝑁 ∈ 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.
StepHypRef Expression
1 ovex 7390 . . . . 5 (𝑁 WWalksN 𝐺) ∈ V
21mptrabex 7175 . . . 4 (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 prefix 𝑁)) ∈ V
32resex 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)}
65, 4clwwlkf1o 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))
87eqeq1d 2738 . . . . . . . 8 (𝑦 = (𝑤 prefix 𝑁) → ((𝑦‘0) = 𝑋 ↔ ((𝑤 prefix 𝑁)‘0) = 𝑋))
983ad2ant3 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))
1210, 11eqeq12d 2752 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ((lastS‘𝑥) = (𝑥‘0) ↔ (lastS‘𝑤) = (𝑤‘0)))
1312elrab 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)
1614, 15wwlknp 28788 . . . . . . . . . . . . . 14 (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑤𝑖), (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 ∈ (ℤ𝑁))
20 peano2uz 12826 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ𝑁) → (𝑁 + 1) ∈ (ℤ𝑁))
2118, 19, 203syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (ℤ𝑁))
22 elfz1end 13471 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
2322biimpi 215 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁))
24 fzss2 13481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 + 1) ∈ (ℤ𝑁) → (1...𝑁) ⊆ (1...(𝑁 + 1)))
2524sselda 3944 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 + 1) ∈ (ℤ𝑁) ∧ 𝑁 ∈ (1...𝑁)) → 𝑁 ∈ (1...(𝑁 + 1)))
2621, 23, 25syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
2726adantl 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)))
2928eleq2d 2823 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑤) = (𝑁 + 1) → (𝑁 ∈ (1...(♯‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3029adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) → (𝑁 ∈ (1...(♯‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3130adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ (1...(♯‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3227, 31mpbird 256 . . . . . . . . . . . . . . . . 17 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(♯‘𝑤)))
3317, 32jca 512 . . . . . . . . . . . . . . . 16 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤))))
3433ex 413 . . . . . . . . . . . . . . 15 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
35343adant3 1132 . . . . . . . . . . . . . 14 ((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 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
3736adantr 481 . . . . . . . . . . . 12 ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
3813, 37sylbi 216 . . . . . . . . . . 11 (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
3938impcom 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))
4139, 40syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)}) → ((𝑤 prefix 𝑁)‘0) = (𝑤‘0))
4241eqeq1d 2738 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)}) → (((𝑤 prefix 𝑁)‘0) = 𝑋 ↔ (𝑤‘0) = 𝑋))
43423adant3 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))
449, 43bitrd 278 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ∧ 𝑦 = (𝑤 prefix 𝑁)) → ((𝑦‘0) = 𝑋 ↔ (𝑤‘0) = 𝑋))
454, 6, 44f1oresrab 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})
4645adantl 482 . . . 4 ((𝑋𝑉𝑁 ∈ 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}
4847a1i 11 . . . . 5 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑋})
4948f1oeq3d 6781 . . . 4 ((𝑋𝑉𝑁 ∈ 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}))
5046, 49mpbird 256 . . 3 ((𝑋𝑉𝑁 ∈ 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)))
5251spcegv 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)))
533, 50, 52mpsyl 68 . 2 ((𝑋𝑉𝑁 ∈ 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)))
5655bicomi 223 . . . . . 6 ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)) ↔ ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) ∧ (𝑤‘0) = 𝑋))
5756abbii 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)}
5813bicomi 223 . . . . . . . 8 ((𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)) ↔ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)})
5958anbi1i 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))
6059abbii 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)}
6260, 61eqtr4i 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}
6354, 57, 623eqtri 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)))
6563, 64mp1i 13 . . 3 ((𝑋𝑉𝑁 ∈ 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)))
6665exbidv 1924 . 2 ((𝑋𝑉𝑁 ∈ 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)))
6753, 66mpbird 256 1 ((𝑋𝑉𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
Colors of variables: wff setvar class
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-ontowf1o 6495  cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052   + caddc 11054  cn 12153  cz 12499  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
  Copyright terms: Public domain W3C validator