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Theorem clwwlkvbij 29099
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 ∈ β„•) β†’ βˆƒπ‘“ 𝑓:{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))
Distinct variable groups:   𝑓,𝐺,𝑀   𝑓,𝑁,𝑀   𝑓,𝑉   𝑓,𝑋,𝑀
Allowed substitution hint:   𝑉(𝑀)

Proof of Theorem clwwlkvbij
Dummy variables π‘₯ 𝑦 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 7391 . . . . 5 (𝑁 WWalksN 𝐺) ∈ V
21mptrabex 7176 . . . 4 (𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)) ∈ V
32resex 5986 . . 3 ((𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)) β†Ύ {𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}) ∈ V
4 eqid 2733 . . . . . 6 (𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)) = (𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁))
5 eqid 2733 . . . . . . 7 {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} = {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)}
65, 4clwwlkf1o 29037 . . . . . 6 (𝑁 ∈ β„• β†’ (𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)):{π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)}–1-1-ontoβ†’(𝑁 ClWWalksN 𝐺))
7 fveq1 6842 . . . . . . . . 9 (𝑦 = (𝑀 prefix 𝑁) β†’ (π‘¦β€˜0) = ((𝑀 prefix 𝑁)β€˜0))
87eqeq1d 2735 . . . . . . . 8 (𝑦 = (𝑀 prefix 𝑁) β†’ ((π‘¦β€˜0) = 𝑋 ↔ ((𝑀 prefix 𝑁)β€˜0) = 𝑋))
983ad2ant3 1136 . . . . . . 7 ((𝑁 ∈ β„• ∧ 𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∧ 𝑦 = (𝑀 prefix 𝑁)) β†’ ((π‘¦β€˜0) = 𝑋 ↔ ((𝑀 prefix 𝑁)β€˜0) = 𝑋))
10 fveq2 6843 . . . . . . . . . . . . . 14 (π‘₯ = 𝑀 β†’ (lastSβ€˜π‘₯) = (lastSβ€˜π‘€))
11 fveq1 6842 . . . . . . . . . . . . . 14 (π‘₯ = 𝑀 β†’ (π‘₯β€˜0) = (π‘€β€˜0))
1210, 11eqeq12d 2749 . . . . . . . . . . . . 13 (π‘₯ = 𝑀 β†’ ((lastSβ€˜π‘₯) = (π‘₯β€˜0) ↔ (lastSβ€˜π‘€) = (π‘€β€˜0)))
1312elrab 3646 . . . . . . . . . . . 12 (𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↔ (𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘€) = (π‘€β€˜0)))
14 eqid 2733 . . . . . . . . . . . . . . 15 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
15 eqid 2733 . . . . . . . . . . . . . . 15 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
1614, 15wwlknp 28830 . . . . . . . . . . . . . 14 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ (𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
17 simpll 766 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1)) ∧ 𝑁 ∈ β„•) β†’ 𝑀 ∈ Word (Vtxβ€˜πΊ))
18 nnz 12525 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„€)
19 uzid 12783 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„€ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘))
20 peano2uz 12831 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (β„€β‰₯β€˜π‘) β†’ (𝑁 + 1) ∈ (β„€β‰₯β€˜π‘))
2118, 19, 203syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„• β†’ (𝑁 + 1) ∈ (β„€β‰₯β€˜π‘))
22 elfz1end 13477 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„• ↔ 𝑁 ∈ (1...𝑁))
2322biimpi 215 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„• β†’ 𝑁 ∈ (1...𝑁))
24 fzss2 13487 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 + 1) ∈ (β„€β‰₯β€˜π‘) β†’ (1...𝑁) βŠ† (1...(𝑁 + 1)))
2524sselda 3945 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 + 1) ∈ (β„€β‰₯β€˜π‘) ∧ 𝑁 ∈ (1...𝑁)) β†’ 𝑁 ∈ (1...(𝑁 + 1)))
2621, 23, 25syl2anc 585 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„• β†’ 𝑁 ∈ (1...(𝑁 + 1)))
2726adantl 483 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1)) ∧ 𝑁 ∈ β„•) β†’ 𝑁 ∈ (1...(𝑁 + 1)))
28 oveq2 7366 . . . . . . . . . . . . . . . . . . . . 21 ((β™―β€˜π‘€) = (𝑁 + 1) β†’ (1...(β™―β€˜π‘€)) = (1...(𝑁 + 1)))
2928eleq2d 2820 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘€) = (𝑁 + 1) β†’ (𝑁 ∈ (1...(β™―β€˜π‘€)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3029adantl 483 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1)) β†’ (𝑁 ∈ (1...(β™―β€˜π‘€)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3130adantr 482 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1)) ∧ 𝑁 ∈ β„•) β†’ (𝑁 ∈ (1...(β™―β€˜π‘€)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3227, 31mpbird 257 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1)) ∧ 𝑁 ∈ β„•) β†’ 𝑁 ∈ (1...(β™―β€˜π‘€)))
3317, 32jca 513 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1)) ∧ 𝑁 ∈ β„•) β†’ (𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (1...(β™―β€˜π‘€))))
3433ex 414 . . . . . . . . . . . . . . 15 ((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1)) β†’ (𝑁 ∈ β„• β†’ (𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (1...(β™―β€˜π‘€)))))
35343adant3 1133 . . . . . . . . . . . . . 14 ((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) β†’ (𝑁 ∈ β„• β†’ (𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (1...(β™―β€˜π‘€)))))
3616, 35syl 17 . . . . . . . . . . . . 13 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ (𝑁 ∈ β„• β†’ (𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (1...(β™―β€˜π‘€)))))
3736adantr 482 . . . . . . . . . . . 12 ((𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘€) = (π‘€β€˜0)) β†’ (𝑁 ∈ β„• β†’ (𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (1...(β™―β€˜π‘€)))))
3813, 37sylbi 216 . . . . . . . . . . 11 (𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} β†’ (𝑁 ∈ β„• β†’ (𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (1...(β™―β€˜π‘€)))))
3938impcom 409 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ 𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)}) β†’ (𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (1...(β™―β€˜π‘€))))
40 pfxfv0 14586 . . . . . . . . . 10 ((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (1...(β™―β€˜π‘€))) β†’ ((𝑀 prefix 𝑁)β€˜0) = (π‘€β€˜0))
4139, 40syl 17 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ 𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)}) β†’ ((𝑀 prefix 𝑁)β€˜0) = (π‘€β€˜0))
4241eqeq1d 2735 . . . . . . . 8 ((𝑁 ∈ β„• ∧ 𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)}) β†’ (((𝑀 prefix 𝑁)β€˜0) = 𝑋 ↔ (π‘€β€˜0) = 𝑋))
43423adant3 1133 . . . . . . 7 ((𝑁 ∈ β„• ∧ 𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∧ 𝑦 = (𝑀 prefix 𝑁)) β†’ (((𝑀 prefix 𝑁)β€˜0) = 𝑋 ↔ (π‘€β€˜0) = 𝑋))
449, 43bitrd 279 . . . . . 6 ((𝑁 ∈ β„• ∧ 𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∧ 𝑦 = (𝑀 prefix 𝑁)) β†’ ((π‘¦β€˜0) = 𝑋 ↔ (π‘€β€˜0) = 𝑋))
454, 6, 44f1oresrab 7074 . . . . 5 (𝑁 ∈ β„• β†’ ((𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)) β†Ύ {𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}):{𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}–1-1-ontoβ†’{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘¦β€˜0) = 𝑋})
4645adantl 483 . . . 4 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ ((𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)) β†Ύ {𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}):{𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}–1-1-ontoβ†’{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘¦β€˜0) = 𝑋})
47 clwwlknon 29076 . . . . . 6 (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) = {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘¦β€˜0) = 𝑋}
4847a1i 11 . . . . 5 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) = {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘¦β€˜0) = 𝑋})
4948f1oeq3d 6782 . . . 4 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (((𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)) β†Ύ {𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}):{𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ↔ ((𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)) β†Ύ {𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}):{𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}–1-1-ontoβ†’{𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘¦β€˜0) = 𝑋}))
5046, 49mpbird 257 . . 3 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ ((𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)) β†Ύ {𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}):{𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))
51 f1oeq1 6773 . . . 4 (𝑓 = ((𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)) β†Ύ {𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}) β†’ (𝑓:{𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ↔ ((𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)) β†Ύ {𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}):{𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁)))
5251spcegv 3555 . . 3 (((𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)) β†Ύ {𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}) ∈ V β†’ (((𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)) β†Ύ {𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}):{𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁) β†’ βˆƒπ‘“ 𝑓:{𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁)))
533, 50, 52mpsyl 68 . 2 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ βˆƒπ‘“ 𝑓:{𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))
54 df-rab 3407 . . . . 5 {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)} = {𝑀 ∣ (𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋))}
55 anass 470 . . . . . . 7 (((𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘€) = (π‘€β€˜0)) ∧ (π‘€β€˜0) = 𝑋) ↔ (𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)))
5655bicomi 223 . . . . . 6 ((𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)) ↔ ((𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘€) = (π‘€β€˜0)) ∧ (π‘€β€˜0) = 𝑋))
5756abbii 2803 . . . . 5 {𝑀 ∣ (𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋))} = {𝑀 ∣ ((𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘€) = (π‘€β€˜0)) ∧ (π‘€β€˜0) = 𝑋)}
5813bicomi 223 . . . . . . . 8 ((𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘€) = (π‘€β€˜0)) ↔ 𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)})
5958anbi1i 625 . . . . . . 7 (((𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘€) = (π‘€β€˜0)) ∧ (π‘€β€˜0) = 𝑋) ↔ (𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∧ (π‘€β€˜0) = 𝑋))
6059abbii 2803 . . . . . 6 {𝑀 ∣ ((𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘€) = (π‘€β€˜0)) ∧ (π‘€β€˜0) = 𝑋)} = {𝑀 ∣ (𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∧ (π‘€β€˜0) = 𝑋)}
61 df-rab 3407 . . . . . 6 {𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋} = {𝑀 ∣ (𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∧ (π‘€β€˜0) = 𝑋)}
6260, 61eqtr4i 2764 . . . . 5 {𝑀 ∣ ((𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘€) = (π‘€β€˜0)) ∧ (π‘€β€˜0) = 𝑋)} = {𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}
6354, 57, 623eqtri 2765 . . . 4 {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)} = {𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}
64 f1oeq2 6774 . . . 4 ({𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)} = {𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋} β†’ (𝑓:{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ↔ 𝑓:{𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁)))
6563, 64mp1i 13 . . 3 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑓:{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ↔ 𝑓:{𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁)))
6665exbidv 1925 . 2 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (βˆƒπ‘“ 𝑓:{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ↔ βˆƒπ‘“ 𝑓:{𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ∣ (π‘€β€˜0) = 𝑋}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁)))
6753, 66mpbird 257 1 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ βˆƒπ‘“ 𝑓:{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastSβ€˜π‘€) = (π‘€β€˜0) ∧ (π‘€β€˜0) = 𝑋)}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  {crab 3406  Vcvv 3444  {cpr 4589   ↦ cmpt 5189   β†Ύ cres 5636  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  0cc0 11056  1c1 11057   + caddc 11059  β„•cn 12158  β„€cz 12504  β„€β‰₯cuz 12768  ...cfz 13430  ..^cfzo 13573  β™―chash 14236  Word cword 14408  lastSclsw 14456   prefix cpfx 14564  Vtxcvtx 27989  Edgcedg 28040   WWalksN cwwlksn 28813   ClWWalksN cclwwlkn 29010  ClWWalksNOncclwwlknon 29073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-oadd 8417  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-n0 12419  df-xnn0 12491  df-z 12505  df-uz 12769  df-rp 12921  df-fz 13431  df-fzo 13574  df-hash 14237  df-word 14409  df-lsw 14457  df-concat 14465  df-s1 14490  df-substr 14535  df-pfx 14565  df-wwlks 28817  df-wwlksn 28818  df-clwwlk 28968  df-clwwlkn 29011  df-clwwlknon 29074
This theorem is referenced by:  numclwwlkqhash  29361
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