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Theorem wlkl0 29353
Description: There is exactly one walk of length 0 on each vertex 𝑋. (Contributed by AV, 4-Jun-2022.)
Hypothesis
Ref Expression
clwlknon2num.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
wlkl0 (𝑋 ∈ 𝑉 β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} = {βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩})
Distinct variable groups:   𝑀,𝐺   𝑀,𝑉   𝑀,𝑋

Proof of Theorem wlkl0
StepHypRef Expression
1 clwlkwlk 28765 . . . . . . . 8 (𝑀 ∈ (ClWalksβ€˜πΊ) β†’ 𝑀 ∈ (Walksβ€˜πΊ))
2 wlkop 28618 . . . . . . . 8 (𝑀 ∈ (Walksβ€˜πΊ) β†’ 𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
31, 2syl 17 . . . . . . 7 (𝑀 ∈ (ClWalksβ€˜πΊ) β†’ 𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
4 fvex 6860 . . . . . . . . . . . . . . 15 (1st β€˜π‘€) ∈ V
5 hasheq0 14270 . . . . . . . . . . . . . . 15 ((1st β€˜π‘€) ∈ V β†’ ((β™―β€˜(1st β€˜π‘€)) = 0 ↔ (1st β€˜π‘€) = βˆ…))
64, 5ax-mp 5 . . . . . . . . . . . . . 14 ((β™―β€˜(1st β€˜π‘€)) = 0 ↔ (1st β€˜π‘€) = βˆ…)
76biimpi 215 . . . . . . . . . . . . 13 ((β™―β€˜(1st β€˜π‘€)) = 0 β†’ (1st β€˜π‘€) = βˆ…)
87adantr 482 . . . . . . . . . . . 12 (((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) β†’ (1st β€˜π‘€) = βˆ…)
983ad2ant3 1136 . . . . . . . . . . 11 ((𝑋 ∈ 𝑉 ∧ (1st β€˜π‘€)(ClWalksβ€˜πΊ)(2nd β€˜π‘€) ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) β†’ (1st β€˜π‘€) = βˆ…)
108adantl 483 . . . . . . . . . . . . . . . . 17 ((𝑋 ∈ 𝑉 ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) β†’ (1st β€˜π‘€) = βˆ…)
1110breq1d 5120 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ 𝑉 ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) β†’ ((1st β€˜π‘€)(ClWalksβ€˜πΊ)(2nd β€˜π‘€) ↔ βˆ…(ClWalksβ€˜πΊ)(2nd β€˜π‘€)))
12 clwlknon2num.v . . . . . . . . . . . . . . . . . . 19 𝑉 = (Vtxβ€˜πΊ)
13121vgrex 27995 . . . . . . . . . . . . . . . . . 18 (𝑋 ∈ 𝑉 β†’ 𝐺 ∈ V)
14120clwlk 29116 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ V β†’ (βˆ…(ClWalksβ€˜πΊ)(2nd β€˜π‘€) ↔ (2nd β€˜π‘€):(0...0)βŸΆπ‘‰))
1513, 14syl 17 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ 𝑉 β†’ (βˆ…(ClWalksβ€˜πΊ)(2nd β€˜π‘€) ↔ (2nd β€˜π‘€):(0...0)βŸΆπ‘‰))
1615adantr 482 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ 𝑉 ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) β†’ (βˆ…(ClWalksβ€˜πΊ)(2nd β€˜π‘€) ↔ (2nd β€˜π‘€):(0...0)βŸΆπ‘‰))
1711, 16bitrd 279 . . . . . . . . . . . . . . 15 ((𝑋 ∈ 𝑉 ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) β†’ ((1st β€˜π‘€)(ClWalksβ€˜πΊ)(2nd β€˜π‘€) ↔ (2nd β€˜π‘€):(0...0)βŸΆπ‘‰))
18 fz0sn 13548 . . . . . . . . . . . . . . . . 17 (0...0) = {0}
1918feq2i 6665 . . . . . . . . . . . . . . . 16 ((2nd β€˜π‘€):(0...0)βŸΆπ‘‰ ↔ (2nd β€˜π‘€):{0}βŸΆπ‘‰)
20 c0ex 11156 . . . . . . . . . . . . . . . . . 18 0 ∈ V
2120fsn2 7087 . . . . . . . . . . . . . . . . 17 ((2nd β€˜π‘€):{0}βŸΆπ‘‰ ↔ (((2nd β€˜π‘€)β€˜0) ∈ 𝑉 ∧ (2nd β€˜π‘€) = {⟨0, ((2nd β€˜π‘€)β€˜0)⟩}))
22 simprr 772 . . . . . . . . . . . . . . . . . . 19 (((𝑋 ∈ 𝑉 ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) ∧ (((2nd β€˜π‘€)β€˜0) ∈ 𝑉 ∧ (2nd β€˜π‘€) = {⟨0, ((2nd β€˜π‘€)β€˜0)⟩})) β†’ (2nd β€˜π‘€) = {⟨0, ((2nd β€˜π‘€)β€˜0)⟩})
23 simprr 772 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ 𝑉 ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) β†’ ((2nd β€˜π‘€)β€˜0) = 𝑋)
2423adantr 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝑋 ∈ 𝑉 ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) ∧ (((2nd β€˜π‘€)β€˜0) ∈ 𝑉 ∧ (2nd β€˜π‘€) = {⟨0, ((2nd β€˜π‘€)β€˜0)⟩})) β†’ ((2nd β€˜π‘€)β€˜0) = 𝑋)
2524opeq2d 4842 . . . . . . . . . . . . . . . . . . . 20 (((𝑋 ∈ 𝑉 ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) ∧ (((2nd β€˜π‘€)β€˜0) ∈ 𝑉 ∧ (2nd β€˜π‘€) = {⟨0, ((2nd β€˜π‘€)β€˜0)⟩})) β†’ ⟨0, ((2nd β€˜π‘€)β€˜0)⟩ = ⟨0, π‘‹βŸ©)
2625sneqd 4603 . . . . . . . . . . . . . . . . . . 19 (((𝑋 ∈ 𝑉 ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) ∧ (((2nd β€˜π‘€)β€˜0) ∈ 𝑉 ∧ (2nd β€˜π‘€) = {⟨0, ((2nd β€˜π‘€)β€˜0)⟩})) β†’ {⟨0, ((2nd β€˜π‘€)β€˜0)⟩} = {⟨0, π‘‹βŸ©})
2722, 26eqtrd 2777 . . . . . . . . . . . . . . . . . 18 (((𝑋 ∈ 𝑉 ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) ∧ (((2nd β€˜π‘€)β€˜0) ∈ 𝑉 ∧ (2nd β€˜π‘€) = {⟨0, ((2nd β€˜π‘€)β€˜0)⟩})) β†’ (2nd β€˜π‘€) = {⟨0, π‘‹βŸ©})
2827ex 414 . . . . . . . . . . . . . . . . 17 ((𝑋 ∈ 𝑉 ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) β†’ ((((2nd β€˜π‘€)β€˜0) ∈ 𝑉 ∧ (2nd β€˜π‘€) = {⟨0, ((2nd β€˜π‘€)β€˜0)⟩}) β†’ (2nd β€˜π‘€) = {⟨0, π‘‹βŸ©}))
2921, 28biimtrid 241 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ 𝑉 ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) β†’ ((2nd β€˜π‘€):{0}βŸΆπ‘‰ β†’ (2nd β€˜π‘€) = {⟨0, π‘‹βŸ©}))
3019, 29biimtrid 241 . . . . . . . . . . . . . . 15 ((𝑋 ∈ 𝑉 ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) β†’ ((2nd β€˜π‘€):(0...0)βŸΆπ‘‰ β†’ (2nd β€˜π‘€) = {⟨0, π‘‹βŸ©}))
3117, 30sylbid 239 . . . . . . . . . . . . . 14 ((𝑋 ∈ 𝑉 ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) β†’ ((1st β€˜π‘€)(ClWalksβ€˜πΊ)(2nd β€˜π‘€) β†’ (2nd β€˜π‘€) = {⟨0, π‘‹βŸ©}))
3231ex 414 . . . . . . . . . . . . 13 (𝑋 ∈ 𝑉 β†’ (((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) β†’ ((1st β€˜π‘€)(ClWalksβ€˜πΊ)(2nd β€˜π‘€) β†’ (2nd β€˜π‘€) = {⟨0, π‘‹βŸ©})))
3332com23 86 . . . . . . . . . . . 12 (𝑋 ∈ 𝑉 β†’ ((1st β€˜π‘€)(ClWalksβ€˜πΊ)(2nd β€˜π‘€) β†’ (((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) β†’ (2nd β€˜π‘€) = {⟨0, π‘‹βŸ©})))
34333imp 1112 . . . . . . . . . . 11 ((𝑋 ∈ 𝑉 ∧ (1st β€˜π‘€)(ClWalksβ€˜πΊ)(2nd β€˜π‘€) ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) β†’ (2nd β€˜π‘€) = {⟨0, π‘‹βŸ©})
359, 34opeq12d 4843 . . . . . . . . . 10 ((𝑋 ∈ 𝑉 ∧ (1st β€˜π‘€)(ClWalksβ€˜πΊ)(2nd β€˜π‘€) ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) β†’ ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩)
36353exp 1120 . . . . . . . . 9 (𝑋 ∈ 𝑉 β†’ ((1st β€˜π‘€)(ClWalksβ€˜πΊ)(2nd β€˜π‘€) β†’ (((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) β†’ ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩)))
37 eleq1 2826 . . . . . . . . . . 11 (𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ β†’ (𝑀 ∈ (ClWalksβ€˜πΊ) ↔ ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ ∈ (ClWalksβ€˜πΊ)))
38 df-br 5111 . . . . . . . . . . 11 ((1st β€˜π‘€)(ClWalksβ€˜πΊ)(2nd β€˜π‘€) ↔ ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ ∈ (ClWalksβ€˜πΊ))
3937, 38bitr4di 289 . . . . . . . . . 10 (𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ β†’ (𝑀 ∈ (ClWalksβ€˜πΊ) ↔ (1st β€˜π‘€)(ClWalksβ€˜πΊ)(2nd β€˜π‘€)))
40 eqeq1 2741 . . . . . . . . . . 11 (𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ β†’ (𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ ↔ ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩))
4140imbi2d 341 . . . . . . . . . 10 (𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ β†’ ((((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) β†’ 𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩) ↔ (((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) β†’ ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩)))
4239, 41imbi12d 345 . . . . . . . . 9 (𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ β†’ ((𝑀 ∈ (ClWalksβ€˜πΊ) β†’ (((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) β†’ 𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩)) ↔ ((1st β€˜π‘€)(ClWalksβ€˜πΊ)(2nd β€˜π‘€) β†’ (((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) β†’ ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩))))
4336, 42syl5ibr 246 . . . . . . . 8 (𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ β†’ (𝑋 ∈ 𝑉 β†’ (𝑀 ∈ (ClWalksβ€˜πΊ) β†’ (((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) β†’ 𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩))))
4443com23 86 . . . . . . 7 (𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ β†’ (𝑀 ∈ (ClWalksβ€˜πΊ) β†’ (𝑋 ∈ 𝑉 β†’ (((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) β†’ 𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩))))
453, 44mpcom 38 . . . . . 6 (𝑀 ∈ (ClWalksβ€˜πΊ) β†’ (𝑋 ∈ 𝑉 β†’ (((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) β†’ 𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩)))
4645com12 32 . . . . 5 (𝑋 ∈ 𝑉 β†’ (𝑀 ∈ (ClWalksβ€˜πΊ) β†’ (((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) β†’ 𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩)))
4746impd 412 . . . 4 (𝑋 ∈ 𝑉 β†’ ((𝑀 ∈ (ClWalksβ€˜πΊ) ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) β†’ 𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩))
48 eqidd 2738 . . . . . . 7 (𝑋 ∈ 𝑉 β†’ βˆ… = βˆ…)
4920a1i 11 . . . . . . . 8 (𝑋 ∈ 𝑉 β†’ 0 ∈ V)
50 snidg 4625 . . . . . . . 8 (𝑋 ∈ 𝑉 β†’ 𝑋 ∈ {𝑋})
5149, 50fsnd 6832 . . . . . . 7 (𝑋 ∈ 𝑉 β†’ {⟨0, π‘‹βŸ©}:{0}⟢{𝑋})
52120clwlkv 29117 . . . . . . 7 ((𝑋 ∈ 𝑉 ∧ βˆ… = βˆ… ∧ {⟨0, π‘‹βŸ©}:{0}⟢{𝑋}) β†’ βˆ…(ClWalksβ€˜πΊ){⟨0, π‘‹βŸ©})
5348, 51, 52mpd3an23 1464 . . . . . 6 (𝑋 ∈ 𝑉 β†’ βˆ…(ClWalksβ€˜πΊ){⟨0, π‘‹βŸ©})
54 hash0 14274 . . . . . . 7 (β™―β€˜βˆ…) = 0
5554a1i 11 . . . . . 6 (𝑋 ∈ 𝑉 β†’ (β™―β€˜βˆ…) = 0)
56 fvsng 7131 . . . . . . 7 ((0 ∈ V ∧ 𝑋 ∈ 𝑉) β†’ ({⟨0, π‘‹βŸ©}β€˜0) = 𝑋)
5720, 56mpan 689 . . . . . 6 (𝑋 ∈ 𝑉 β†’ ({⟨0, π‘‹βŸ©}β€˜0) = 𝑋)
5853, 55, 57jca32 517 . . . . 5 (𝑋 ∈ 𝑉 β†’ (βˆ…(ClWalksβ€˜πΊ){⟨0, π‘‹βŸ©} ∧ ((β™―β€˜βˆ…) = 0 ∧ ({⟨0, π‘‹βŸ©}β€˜0) = 𝑋)))
59 eleq1 2826 . . . . . . 7 (𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ β†’ (𝑀 ∈ (ClWalksβ€˜πΊ) ↔ βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ ∈ (ClWalksβ€˜πΊ)))
60 df-br 5111 . . . . . . 7 (βˆ…(ClWalksβ€˜πΊ){⟨0, π‘‹βŸ©} ↔ βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ ∈ (ClWalksβ€˜πΊ))
6159, 60bitr4di 289 . . . . . 6 (𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ β†’ (𝑀 ∈ (ClWalksβ€˜πΊ) ↔ βˆ…(ClWalksβ€˜πΊ){⟨0, π‘‹βŸ©}))
62 0ex 5269 . . . . . . . . 9 βˆ… ∈ V
63 snex 5393 . . . . . . . . 9 {⟨0, π‘‹βŸ©} ∈ V
6462, 63op1std 7936 . . . . . . . 8 (𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ β†’ (1st β€˜π‘€) = βˆ…)
6564fveqeq2d 6855 . . . . . . 7 (𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ β†’ ((β™―β€˜(1st β€˜π‘€)) = 0 ↔ (β™―β€˜βˆ…) = 0))
6662, 63op2ndd 7937 . . . . . . . . 9 (𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ β†’ (2nd β€˜π‘€) = {⟨0, π‘‹βŸ©})
6766fveq1d 6849 . . . . . . . 8 (𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ β†’ ((2nd β€˜π‘€)β€˜0) = ({⟨0, π‘‹βŸ©}β€˜0))
6867eqeq1d 2739 . . . . . . 7 (𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ β†’ (((2nd β€˜π‘€)β€˜0) = 𝑋 ↔ ({⟨0, π‘‹βŸ©}β€˜0) = 𝑋))
6965, 68anbi12d 632 . . . . . 6 (𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ β†’ (((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) ↔ ((β™―β€˜βˆ…) = 0 ∧ ({⟨0, π‘‹βŸ©}β€˜0) = 𝑋)))
7061, 69anbi12d 632 . . . . 5 (𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ β†’ ((𝑀 ∈ (ClWalksβ€˜πΊ) ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) ↔ (βˆ…(ClWalksβ€˜πΊ){⟨0, π‘‹βŸ©} ∧ ((β™―β€˜βˆ…) = 0 ∧ ({⟨0, π‘‹βŸ©}β€˜0) = 𝑋))))
7158, 70syl5ibrcom 247 . . . 4 (𝑋 ∈ 𝑉 β†’ (𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ β†’ (𝑀 ∈ (ClWalksβ€˜πΊ) ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋))))
7247, 71impbid 211 . . 3 (𝑋 ∈ 𝑉 β†’ ((𝑀 ∈ (ClWalksβ€˜πΊ) ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) ↔ 𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩))
7372alrimiv 1931 . 2 (𝑋 ∈ 𝑉 β†’ βˆ€π‘€((𝑀 ∈ (ClWalksβ€˜πΊ) ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) ↔ 𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩))
74 rabeqsn 4632 . 2 ({𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} = {βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩} ↔ βˆ€π‘€((𝑀 ∈ (ClWalksβ€˜πΊ) ∧ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) ↔ 𝑀 = βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩))
7573, 74sylibr 233 1 (𝑋 ∈ 𝑉 β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} = {βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  {crab 3410  Vcvv 3448  βˆ…c0 4287  {csn 4591  βŸ¨cop 4597   class class class wbr 5110  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  1st c1st 7924  2nd c2nd 7925  0cc0 11058  ...cfz 13431  β™―chash 14237  Vtxcvtx 27989  Walkscwlks 28586  ClWalkscclwlks 28760
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-wlks 28589  df-clwlks 28761
This theorem is referenced by:  numclwlk1lem1  29355
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