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Theorem clwwlkvbij 29965
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 7448 . . . . 5 (𝑁 WWalksN 𝐺) ∈ V
21mptrabex 7232 . . . 4 (𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)) ∈ V
32resex 6028 . . 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 2725 . . . . . 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 2725 . . . . . . 7 {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} = {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)}
65, 4clwwlkf1o 29903 . . . . . 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 6890 . . . . . . . . 9 (𝑦 = (𝑀 prefix 𝑁) β†’ (π‘¦β€˜0) = ((𝑀 prefix 𝑁)β€˜0))
87eqeq1d 2727 . . . . . . . 8 (𝑦 = (𝑀 prefix 𝑁) β†’ ((π‘¦β€˜0) = 𝑋 ↔ ((𝑀 prefix 𝑁)β€˜0) = 𝑋))
983ad2ant3 1132 . . . . . . 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 6891 . . . . . . . . . . . . . 14 (π‘₯ = 𝑀 β†’ (lastSβ€˜π‘₯) = (lastSβ€˜π‘€))
11 fveq1 6890 . . . . . . . . . . . . . 14 (π‘₯ = 𝑀 β†’ (π‘₯β€˜0) = (π‘€β€˜0))
1210, 11eqeq12d 2741 . . . . . . . . . . . . 13 (π‘₯ = 𝑀 β†’ ((lastSβ€˜π‘₯) = (π‘₯β€˜0) ↔ (lastSβ€˜π‘€) = (π‘€β€˜0)))
1312elrab 3675 . . . . . . . . . . . 12 (𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↔ (𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘€) = (π‘€β€˜0)))
14 eqid 2725 . . . . . . . . . . . . . . 15 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
15 eqid 2725 . . . . . . . . . . . . . . 15 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
1614, 15wwlknp 29696 . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1)) ∧ 𝑁 ∈ β„•) β†’ 𝑀 ∈ Word (Vtxβ€˜πΊ))
18 nnz 12607 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„€)
19 uzid 12865 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„€ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘))
20 peano2uz 12913 . . . . . . . . . . . . . . . . . . . . 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 13561 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„• ↔ 𝑁 ∈ (1...𝑁))
2322biimpi 215 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„• β†’ 𝑁 ∈ (1...𝑁))
24 fzss2 13571 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 + 1) ∈ (β„€β‰₯β€˜π‘) β†’ (1...𝑁) βŠ† (1...(𝑁 + 1)))
2524sselda 3972 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 + 1) ∈ (β„€β‰₯β€˜π‘) ∧ 𝑁 ∈ (1...𝑁)) β†’ 𝑁 ∈ (1...(𝑁 + 1)))
2621, 23, 25syl2anc 582 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„• β†’ 𝑁 ∈ (1...(𝑁 + 1)))
2726adantl 480 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1)) ∧ 𝑁 ∈ β„•) β†’ 𝑁 ∈ (1...(𝑁 + 1)))
28 oveq2 7423 . . . . . . . . . . . . . . . . . . . . 21 ((β™―β€˜π‘€) = (𝑁 + 1) β†’ (1...(β™―β€˜π‘€)) = (1...(𝑁 + 1)))
2928eleq2d 2811 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘€) = (𝑁 + 1) β†’ (𝑁 ∈ (1...(β™―β€˜π‘€)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3029adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1)) β†’ (𝑁 ∈ (1...(β™―β€˜π‘€)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3130adantr 479 . . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1)) ∧ 𝑁 ∈ β„•) β†’ 𝑁 ∈ (1...(β™―β€˜π‘€)))
3317, 32jca 510 . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . 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 1129 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . 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 406 . . . . . . . . . 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 14672 . . . . . . . . . 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 2727 . . . . . . . 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 1129 . . . . . . 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 278 . . . . . 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 7131 . . . . 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 480 . . . 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 29942 . . . . . 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 6830 . . . 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 256 . . 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 6821 . . . 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 3577 . . 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 3420 . . . . 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 467 . . . . . . 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 2795 . . . . 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 622 . . . . . . 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 2795 . . . . . 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 3420 . . . . . 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 2756 . . . . 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 2757 . . . 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 6822 . . . 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 1916 . 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 256 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 394   ∧ w3a 1084   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  {cab 2702  βˆ€wral 3051  {crab 3419  Vcvv 3463  {cpr 4626   ↦ cmpt 5226   β†Ύ cres 5674  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7415  0cc0 11136  1c1 11137   + caddc 11139  β„•cn 12240  β„€cz 12586  β„€β‰₯cuz 12850  ...cfz 13514  ..^cfzo 13657  β™―chash 14319  Word cword 14494  lastSclsw 14542   prefix cpfx 14650  Vtxcvtx 28851  Edgcedg 28902   WWalksN cwwlksn 29679   ClWWalksN cclwwlkn 29876  ClWWalksNOncclwwlknon 29939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-oadd 8487  df-er 8721  df-map 8843  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-n0 12501  df-xnn0 12573  df-z 12587  df-uz 12851  df-rp 13005  df-fz 13515  df-fzo 13658  df-hash 14320  df-word 14495  df-lsw 14543  df-concat 14551  df-s1 14576  df-substr 14621  df-pfx 14651  df-wwlks 29683  df-wwlksn 29684  df-clwwlk 29834  df-clwwlkn 29877  df-clwwlknon 29940
This theorem is referenced by:  numclwwlkqhash  30227
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