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Theorem clwwlkvbijOLD 27288
Description: Obsolete version of clwwlkvbij 27286 as of 3-Mar-2022. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
clwwlkvbijOLD (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:   𝑓,𝐺,𝑤   𝑓,𝑁,𝑤   𝑆,𝑓,𝑤

Proof of Theorem clwwlkvbijOLD
Dummy variables 𝑥 𝑦 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6822 . . . . . 6 (𝑁 WWalksN 𝐺) ∈ V
21mptrabex 6631 . . . . 5 (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↦ (𝑤 substr ⟨0, 𝑁⟩)) ∈ V
32resex 5584 . . . 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 2770 . . . . 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 2770 . . . . . 6 {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} = {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)}
65, 4clwwlkf1o 27204 . . . . 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 6331 . . . . . . . 8 (𝑦 = (𝑤 substr ⟨0, 𝑁⟩) → (𝑦‘0) = ((𝑤 substr ⟨0, 𝑁⟩)‘0))
87eqeq1d 2772 . . . . . . 7 (𝑦 = (𝑤 substr ⟨0, 𝑁⟩) → ((𝑦‘0) = 𝑆 ↔ ((𝑤 substr ⟨0, 𝑁⟩)‘0) = 𝑆))
983ad2ant3 1128 . . . . . 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 6332 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (lastS‘𝑥) = (lastS‘𝑤))
11 fveq1 6331 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝑥‘0) = (𝑤‘0))
1210, 11eqeq12d 2785 . . . . . . . . . . . 12 (𝑥 = 𝑤 → ((lastS‘𝑥) = (𝑥‘0) ↔ (lastS‘𝑤) = (𝑤‘0)))
1312elrab 3513 . . . . . . . . . . 11 (𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)} ↔ (𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑤) = (𝑤‘0)))
14 eqid 2770 . . . . . . . . . . . . . 14 (Vtx‘𝐺) = (Vtx‘𝐺)
15 eqid 2770 . . . . . . . . . . . . . 14 (Edg‘𝐺) = (Edg‘𝐺)
1614, 15wwlknp 26970 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
17 simpll 742 . . . . . . . . . . . . . . . 16 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑤 ∈ Word (Vtx‘𝐺))
18 nnz 11600 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
19 uzid 11902 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
20 peano2uz 11942 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (ℤ𝑁) → (𝑁 + 1) ∈ (ℤ𝑁))
2118, 19, 203syl 18 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (ℤ𝑁))
22 elfz1end 12577 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
2322biimpi 206 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁))
24 fzss2 12587 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 + 1) ∈ (ℤ𝑁) → (1...𝑁) ⊆ (1...(𝑁 + 1)))
2524sselda 3750 . . . . . . . . . . . . . . . . . . 19 (((𝑁 + 1) ∈ (ℤ𝑁) ∧ 𝑁 ∈ (1...𝑁)) → 𝑁 ∈ (1...(𝑁 + 1)))
2621, 23, 25syl2anc 565 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
2726adantl 467 . . . . . . . . . . . . . . . . 17 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1)))
28 oveq2 6800 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑤) = (𝑁 + 1) → (1...(♯‘𝑤)) = (1...(𝑁 + 1)))
2928eleq2d 2835 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑤) = (𝑁 + 1) → (𝑁 ∈ (1...(♯‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3029adantl 467 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) → (𝑁 ∈ (1...(♯‘𝑤)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3130adantr 466 . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . 15 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤))))
3433ex 397 . . . . . . . . . . . . . 14 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤)))))
35343adant3 1125 . . . . . . . . . . . . 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 466 . . . . . . . . . . 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 394 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)}) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑤))))
40 swrd0fv0 13648 . . . . . . . . 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 2772 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑤 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑥) = (𝑥‘0)}) → (((𝑤 substr ⟨0, 𝑁⟩)‘0) = 𝑆 ↔ (𝑤‘0) = 𝑆))
43423adant3 1125 . . . . . 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 6537 . . . 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 6268 . . . . 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 3443 . . . 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 6331 . . . . . . 7 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
5049eqeq1d 2772 . . . . . 6 (𝑤 = 𝑦 → ((𝑤‘0) = 𝑆 ↔ (𝑦‘0) = 𝑆))
5150cbvrabv 3348 . . . . 5 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆} = {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑦‘0) = 𝑆}
52 f1oeq3 6270 . . . . 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 2001 . . 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 3069 . . . . 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 459 . . . . . . 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 2887 . . . . 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 602 . . . . . . 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 2887 . . . . . 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 3069 . . . . . 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 2795 . . . . 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 2796 . . . 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 6269 . . . 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 2001 . 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 382  w3a 1070   = wceq 1630  wex 1851  wcel 2144  {cab 2756  wral 3060  {crab 3064  Vcvv 3349  {cpr 4316  cop 4320  cmpt 4861  cres 5251  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  0cc0 10137  1c1 10138   + caddc 10140  cn 11221  cz 11578  cuz 11887  ...cfz 12532  ..^cfzo 12672  chash 13320  Word cword 13486  lastSclsw 13487   substr csubstr 13490  Vtxcvtx 26094  Edgcedg 26159   WWalksN cwwlksn 26953   ClWWalksN cclwwlkn 27171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-n0 11494  df-xnn0 11565  df-z 11579  df-uz 11888  df-rp 12035  df-fz 12533  df-fzo 12673  df-hash 13321  df-word 13494  df-lsw 13495  df-concat 13496  df-s1 13497  df-substr 13498  df-wwlks 26957  df-wwlksn 26958  df-clwwlk 27129  df-clwwlkn 27173
This theorem is referenced by: (None)
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