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

Theorem clwwlkvbij 29875
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 7438 . . . . 5 (𝑁 WWalksN 𝐺) ∈ V
21mptrabex 7222 . . . 4 (𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↦ (𝑀 prefix 𝑁)) ∈ V
32resex 6023 . . 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 2726 . . . . . 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 2726 . . . . . . 7 {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} = {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)}
65, 4clwwlkf1o 29813 . . . . . 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 6884 . . . . . . . . 9 (𝑦 = (𝑀 prefix 𝑁) β†’ (π‘¦β€˜0) = ((𝑀 prefix 𝑁)β€˜0))
87eqeq1d 2728 . . . . . . . 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 6885 . . . . . . . . . . . . . 14 (π‘₯ = 𝑀 β†’ (lastSβ€˜π‘₯) = (lastSβ€˜π‘€))
11 fveq1 6884 . . . . . . . . . . . . . 14 (π‘₯ = 𝑀 β†’ (π‘₯β€˜0) = (π‘€β€˜0))
1210, 11eqeq12d 2742 . . . . . . . . . . . . 13 (π‘₯ = 𝑀 β†’ ((lastSβ€˜π‘₯) = (π‘₯β€˜0) ↔ (lastSβ€˜π‘€) = (π‘€β€˜0)))
1312elrab 3678 . . . . . . . . . . . 12 (𝑀 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (lastSβ€˜π‘₯) = (π‘₯β€˜0)} ↔ (𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘€) = (π‘€β€˜0)))
14 eqid 2726 . . . . . . . . . . . . . . 15 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
15 eqid 2726 . . . . . . . . . . . . . . 15 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
1614, 15wwlknp 29606 . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1)) ∧ 𝑁 ∈ β„•) β†’ 𝑀 ∈ Word (Vtxβ€˜πΊ))
18 nnz 12583 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„€)
19 uzid 12841 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„€ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘))
20 peano2uz 12889 . . . . . . . . . . . . . . . . . . . . 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 13537 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„• ↔ 𝑁 ∈ (1...𝑁))
2322biimpi 215 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„• β†’ 𝑁 ∈ (1...𝑁))
24 fzss2 13547 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 + 1) ∈ (β„€β‰₯β€˜π‘) β†’ (1...𝑁) βŠ† (1...(𝑁 + 1)))
2524sselda 3977 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 + 1) ∈ (β„€β‰₯β€˜π‘) ∧ 𝑁 ∈ (1...𝑁)) β†’ 𝑁 ∈ (1...(𝑁 + 1)))
2621, 23, 25syl2anc 583 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„• β†’ 𝑁 ∈ (1...(𝑁 + 1)))
2726adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1)) ∧ 𝑁 ∈ β„•) β†’ 𝑁 ∈ (1...(𝑁 + 1)))
28 oveq2 7413 . . . . . . . . . . . . . . . . . . . . 21 ((β™―β€˜π‘€) = (𝑁 + 1) β†’ (1...(β™―β€˜π‘€)) = (1...(𝑁 + 1)))
2928eleq2d 2813 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘€) = (𝑁 + 1) β†’ (𝑁 ∈ (1...(β™―β€˜π‘€)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3029adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘€) = (𝑁 + 1)) β†’ (𝑁 ∈ (1...(β™―β€˜π‘€)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3130adantr 480 . . . . . . . . . . . . . . . . . 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 511 . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 407 . . . . . . . . . 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 14648 . . . . . . . . . 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 2728 . . . . . . . 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 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 7121 . . . . 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 481 . . . 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 29852 . . . . . 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 6824 . . . 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 6815 . . . 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 3581 . . 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 3427 . . . . 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 468 . . . . . . 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 2796 . . . . 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 623 . . . . . . 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 2796 . . . . . 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 3427 . . . . . 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 2757 . . . . 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 2758 . . . 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 6816 . . . 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 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 395   ∧ w3a 1084   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  {cab 2703  βˆ€wral 3055  {crab 3426  Vcvv 3468  {cpr 4625   ↦ cmpt 5224   β†Ύ cres 5671  β€“1-1-ontoβ†’wf1o 6536  β€˜cfv 6537  (class class class)co 7405  0cc0 11112  1c1 11113   + caddc 11115  β„•cn 12216  β„€cz 12562  β„€β‰₯cuz 12826  ...cfz 13490  ..^cfzo 13633  β™―chash 14295  Word cword 14470  lastSclsw 14518   prefix cpfx 14626  Vtxcvtx 28764  Edgcedg 28815   WWalksN cwwlksn 29589   ClWWalksN cclwwlkn 29786  ClWWalksNOncclwwlknon 29849
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-oadd 8471  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-rp 12981  df-fz 13491  df-fzo 13634  df-hash 14296  df-word 14471  df-lsw 14519  df-concat 14527  df-s1 14552  df-substr 14597  df-pfx 14627  df-wwlks 29593  df-wwlksn 29594  df-clwwlk 29744  df-clwwlkn 29787  df-clwwlknon 29850
This theorem is referenced by:  numclwwlkqhash  30137
  Copyright terms: Public domain W3C validator