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Theorem clwwlknon1 29859
Description: The set of closed walks on vertex 𝑋 of length 1 in a graph 𝐺 as words over the set of vertices. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 24-Mar-2022.)
Hypotheses
Ref Expression
clwwlknon1.v 𝑉 = (Vtxβ€˜πΊ)
clwwlknon1.c 𝐢 = (ClWWalksNOnβ€˜πΊ)
clwwlknon1.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
clwwlknon1 (𝑋 ∈ 𝑉 β†’ (𝑋𝐢1) = {𝑀 ∈ Word 𝑉 ∣ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)})
Distinct variable groups:   𝑀,𝐺   𝑀,𝑉   𝑀,𝑋
Allowed substitution hints:   𝐢(𝑀)   𝐸(𝑀)

Proof of Theorem clwwlknon1
StepHypRef Expression
1 clwwlknon1.c . . . 4 𝐢 = (ClWWalksNOnβ€˜πΊ)
21oveqi 7418 . . 3 (𝑋𝐢1) = (𝑋(ClWWalksNOnβ€˜πΊ)1)
32a1i 11 . 2 (𝑋 ∈ 𝑉 β†’ (𝑋𝐢1) = (𝑋(ClWWalksNOnβ€˜πΊ)1))
4 clwwlknon 29852 . . 3 (𝑋(ClWWalksNOnβ€˜πΊ)1) = {𝑀 ∈ (1 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}
54a1i 11 . 2 (𝑋 ∈ 𝑉 β†’ (𝑋(ClWWalksNOnβ€˜πΊ)1) = {𝑀 ∈ (1 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋})
6 clwwlkn1 29803 . . . . 5 (𝑀 ∈ (1 ClWWalksN 𝐺) ↔ ((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)))
76anbi1i 623 . . . 4 ((𝑀 ∈ (1 ClWWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑋) ↔ (((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (π‘€β€˜0) = 𝑋))
8 clwwlknon1.v . . . . . . . . . . . 12 𝑉 = (Vtxβ€˜πΊ)
98eqcomi 2735 . . . . . . . . . . 11 (Vtxβ€˜πΊ) = 𝑉
109wrdeqi 14493 . . . . . . . . . 10 Word (Vtxβ€˜πΊ) = Word 𝑉
1110eleq2i 2819 . . . . . . . . 9 (𝑀 ∈ Word (Vtxβ€˜πΊ) ↔ 𝑀 ∈ Word 𝑉)
1211biimpi 215 . . . . . . . 8 (𝑀 ∈ Word (Vtxβ€˜πΊ) β†’ 𝑀 ∈ Word 𝑉)
13123ad2ant2 1131 . . . . . . 7 (((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) β†’ 𝑀 ∈ Word 𝑉)
1413ad2antrl 725 . . . . . 6 ((𝑋 ∈ 𝑉 ∧ (((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (π‘€β€˜0) = 𝑋)) β†’ 𝑀 ∈ Word 𝑉)
1513adantr 480 . . . . . . . . 9 ((((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (π‘€β€˜0) = 𝑋) β†’ 𝑀 ∈ Word 𝑉)
16 simpl1 1188 . . . . . . . . 9 ((((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (π‘€β€˜0) = 𝑋) β†’ (β™―β€˜π‘€) = 1)
17 simpr 484 . . . . . . . . 9 ((((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (π‘€β€˜0) = 𝑋) β†’ (π‘€β€˜0) = 𝑋)
1815, 16, 173jca 1125 . . . . . . . 8 ((((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (π‘€β€˜0) = 𝑋) β†’ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = 1 ∧ (π‘€β€˜0) = 𝑋))
1918adantl 481 . . . . . . 7 ((𝑋 ∈ 𝑉 ∧ (((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (π‘€β€˜0) = 𝑋)) β†’ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = 1 ∧ (π‘€β€˜0) = 𝑋))
20 wrdl1s1 14570 . . . . . . . 8 (𝑋 ∈ 𝑉 β†’ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ↔ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = 1 ∧ (π‘€β€˜0) = 𝑋)))
2120adantr 480 . . . . . . 7 ((𝑋 ∈ 𝑉 ∧ (((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (π‘€β€˜0) = 𝑋)) β†’ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ↔ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = 1 ∧ (π‘€β€˜0) = 𝑋)))
2219, 21mpbird 257 . . . . . 6 ((𝑋 ∈ 𝑉 ∧ (((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (π‘€β€˜0) = 𝑋)) β†’ 𝑀 = βŸ¨β€œπ‘‹β€βŸ©)
23 sneq 4633 . . . . . . . . . . . . 13 ((π‘€β€˜0) = 𝑋 β†’ {(π‘€β€˜0)} = {𝑋})
24 clwwlknon1.e . . . . . . . . . . . . . . 15 𝐸 = (Edgβ€˜πΊ)
2524eqcomi 2735 . . . . . . . . . . . . . 14 (Edgβ€˜πΊ) = 𝐸
2625a1i 11 . . . . . . . . . . . . 13 ((π‘€β€˜0) = 𝑋 β†’ (Edgβ€˜πΊ) = 𝐸)
2723, 26eleq12d 2821 . . . . . . . . . . . 12 ((π‘€β€˜0) = 𝑋 β†’ ({(π‘€β€˜0)} ∈ (Edgβ€˜πΊ) ↔ {𝑋} ∈ 𝐸))
2827biimpd 228 . . . . . . . . . . 11 ((π‘€β€˜0) = 𝑋 β†’ ({(π‘€β€˜0)} ∈ (Edgβ€˜πΊ) β†’ {𝑋} ∈ 𝐸))
2928a1i 11 . . . . . . . . . 10 (𝑋 ∈ 𝑉 β†’ ((π‘€β€˜0) = 𝑋 β†’ ({(π‘€β€˜0)} ∈ (Edgβ€˜πΊ) β†’ {𝑋} ∈ 𝐸)))
3029com13 88 . . . . . . . . 9 ({(π‘€β€˜0)} ∈ (Edgβ€˜πΊ) β†’ ((π‘€β€˜0) = 𝑋 β†’ (𝑋 ∈ 𝑉 β†’ {𝑋} ∈ 𝐸)))
31303ad2ant3 1132 . . . . . . . 8 (((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) β†’ ((π‘€β€˜0) = 𝑋 β†’ (𝑋 ∈ 𝑉 β†’ {𝑋} ∈ 𝐸)))
3231imp 406 . . . . . . 7 ((((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (π‘€β€˜0) = 𝑋) β†’ (𝑋 ∈ 𝑉 β†’ {𝑋} ∈ 𝐸))
3332impcom 407 . . . . . 6 ((𝑋 ∈ 𝑉 ∧ (((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (π‘€β€˜0) = 𝑋)) β†’ {𝑋} ∈ 𝐸)
3414, 22, 33jca32 515 . . . . 5 ((𝑋 ∈ 𝑉 ∧ (((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (π‘€β€˜0) = 𝑋)) β†’ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)))
35 fveq2 6885 . . . . . . . . . 10 (𝑀 = βŸ¨β€œπ‘‹β€βŸ© β†’ (β™―β€˜π‘€) = (β™―β€˜βŸ¨β€œπ‘‹β€βŸ©))
36 s1len 14562 . . . . . . . . . 10 (β™―β€˜βŸ¨β€œπ‘‹β€βŸ©) = 1
3735, 36eqtrdi 2782 . . . . . . . . 9 (𝑀 = βŸ¨β€œπ‘‹β€βŸ© β†’ (β™―β€˜π‘€) = 1)
3837ad2antrl 725 . . . . . . . 8 ((𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)) β†’ (β™―β€˜π‘€) = 1)
3938adantl 481 . . . . . . 7 ((𝑋 ∈ 𝑉 ∧ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸))) β†’ (β™―β€˜π‘€) = 1)
408wrdeqi 14493 . . . . . . . . . 10 Word 𝑉 = Word (Vtxβ€˜πΊ)
4140eleq2i 2819 . . . . . . . . 9 (𝑀 ∈ Word 𝑉 ↔ 𝑀 ∈ Word (Vtxβ€˜πΊ))
4241biimpi 215 . . . . . . . 8 (𝑀 ∈ Word 𝑉 β†’ 𝑀 ∈ Word (Vtxβ€˜πΊ))
4342ad2antrl 725 . . . . . . 7 ((𝑋 ∈ 𝑉 ∧ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸))) β†’ 𝑀 ∈ Word (Vtxβ€˜πΊ))
44 fveq1 6884 . . . . . . . . . . . . . . 15 (𝑀 = βŸ¨β€œπ‘‹β€βŸ© β†’ (π‘€β€˜0) = (βŸ¨β€œπ‘‹β€βŸ©β€˜0))
45 s1fv 14566 . . . . . . . . . . . . . . 15 (𝑋 ∈ 𝑉 β†’ (βŸ¨β€œπ‘‹β€βŸ©β€˜0) = 𝑋)
4644, 45sylan9eq 2786 . . . . . . . . . . . . . 14 ((𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ 𝑋 ∈ 𝑉) β†’ (π‘€β€˜0) = 𝑋)
4746eqcomd 2732 . . . . . . . . . . . . 13 ((𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 = (π‘€β€˜0))
4847sneqd 4635 . . . . . . . . . . . 12 ((𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ 𝑋 ∈ 𝑉) β†’ {𝑋} = {(π‘€β€˜0)})
4924a1i 11 . . . . . . . . . . . 12 ((𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ 𝑋 ∈ 𝑉) β†’ 𝐸 = (Edgβ€˜πΊ))
5048, 49eleq12d 2821 . . . . . . . . . . 11 ((𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ 𝑋 ∈ 𝑉) β†’ ({𝑋} ∈ 𝐸 ↔ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)))
5150biimpd 228 . . . . . . . . . 10 ((𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ 𝑋 ∈ 𝑉) β†’ ({𝑋} ∈ 𝐸 β†’ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)))
5251impancom 451 . . . . . . . . 9 ((𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸) β†’ (𝑋 ∈ 𝑉 β†’ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)))
5352adantl 481 . . . . . . . 8 ((𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)) β†’ (𝑋 ∈ 𝑉 β†’ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)))
5453impcom 407 . . . . . . 7 ((𝑋 ∈ 𝑉 ∧ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸))) β†’ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ))
5539, 43, 543jca 1125 . . . . . 6 ((𝑋 ∈ 𝑉 ∧ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸))) β†’ ((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)))
5646ex 412 . . . . . . . 8 (𝑀 = βŸ¨β€œπ‘‹β€βŸ© β†’ (𝑋 ∈ 𝑉 β†’ (π‘€β€˜0) = 𝑋))
5756ad2antrl 725 . . . . . . 7 ((𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)) β†’ (𝑋 ∈ 𝑉 β†’ (π‘€β€˜0) = 𝑋))
5857impcom 407 . . . . . 6 ((𝑋 ∈ 𝑉 ∧ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸))) β†’ (π‘€β€˜0) = 𝑋)
5955, 58jca 511 . . . . 5 ((𝑋 ∈ 𝑉 ∧ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸))) β†’ (((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (π‘€β€˜0) = 𝑋))
6034, 59impbida 798 . . . 4 (𝑋 ∈ 𝑉 β†’ ((((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (π‘€β€˜0) = 𝑋) ↔ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸))))
617, 60bitrid 283 . . 3 (𝑋 ∈ 𝑉 β†’ ((𝑀 ∈ (1 ClWWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑋) ↔ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸))))
6261rabbidva2 3428 . 2 (𝑋 ∈ 𝑉 β†’ {𝑀 ∈ (1 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} = {𝑀 ∈ Word 𝑉 ∣ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)})
633, 5, 623eqtrd 2770 1 (𝑋 ∈ 𝑉 β†’ (𝑋𝐢1) = {𝑀 ∈ Word 𝑉 ∣ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3426  {csn 4623  β€˜cfv 6537  (class class class)co 7405  0cc0 11112  1c1 11113  β™―chash 14295  Word cword 14470  βŸ¨β€œcs1 14551  Vtxcvtx 28764  Edgcedg 28815   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-fz 13491  df-fzo 13634  df-hash 14296  df-word 14471  df-lsw 14519  df-s1 14552  df-clwwlk 29744  df-clwwlkn 29787  df-clwwlknon 29850
This theorem is referenced by:  clwwlknon1loop  29860  clwwlknon1nloop  29861
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