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Theorem clwwlknon1 29949
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 7428 . . 3 (𝑋𝐢1) = (𝑋(ClWWalksNOnβ€˜πΊ)1)
32a1i 11 . 2 (𝑋 ∈ 𝑉 β†’ (𝑋𝐢1) = (𝑋(ClWWalksNOnβ€˜πΊ)1))
4 clwwlknon 29942 . . 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 29893 . . . . 5 (𝑀 ∈ (1 ClWWalksN 𝐺) ↔ ((β™―β€˜π‘€) = 1 ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)))
76anbi1i 622 . . . 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 2734 . . . . . . . . . . 11 (Vtxβ€˜πΊ) = 𝑉
109wrdeqi 14517 . . . . . . . . . 10 Word (Vtxβ€˜πΊ) = Word 𝑉
1110eleq2i 2817 . . . . . . . . 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 726 . . . . . 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 479 . . . . . . . . 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 483 . . . . . . . . 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 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 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = 1 ∧ (π‘€β€˜0) = 𝑋))
20 wrdl1s1 14594 . . . . . . . 8 (𝑋 ∈ 𝑉 β†’ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ↔ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = 1 ∧ (π‘€β€˜0) = 𝑋)))
2120adantr 479 . . . . . . 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 256 . . . . . 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 4634 . . . . . . . . . . . . 13 ((π‘€β€˜0) = 𝑋 β†’ {(π‘€β€˜0)} = {𝑋})
24 clwwlknon1.e . . . . . . . . . . . . . . 15 𝐸 = (Edgβ€˜πΊ)
2524eqcomi 2734 . . . . . . . . . . . . . 14 (Edgβ€˜πΊ) = 𝐸
2625a1i 11 . . . . . . . . . . . . 13 ((π‘€β€˜0) = 𝑋 β†’ (Edgβ€˜πΊ) = 𝐸)
2723, 26eleq12d 2819 . . . . . . . . . . . 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 405 . . . . . . 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 406 . . . . . 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 514 . . . . 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 6891 . . . . . . . . . 10 (𝑀 = βŸ¨β€œπ‘‹β€βŸ© β†’ (β™―β€˜π‘€) = (β™―β€˜βŸ¨β€œπ‘‹β€βŸ©))
36 s1len 14586 . . . . . . . . . 10 (β™―β€˜βŸ¨β€œπ‘‹β€βŸ©) = 1
3735, 36eqtrdi 2781 . . . . . . . . 9 (𝑀 = βŸ¨β€œπ‘‹β€βŸ© β†’ (β™―β€˜π‘€) = 1)
3837ad2antrl 726 . . . . . . . 8 ((𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)) β†’ (β™―β€˜π‘€) = 1)
3938adantl 480 . . . . . . 7 ((𝑋 ∈ 𝑉 ∧ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸))) β†’ (β™―β€˜π‘€) = 1)
408wrdeqi 14517 . . . . . . . . . 10 Word 𝑉 = Word (Vtxβ€˜πΊ)
4140eleq2i 2817 . . . . . . . . 9 (𝑀 ∈ Word 𝑉 ↔ 𝑀 ∈ Word (Vtxβ€˜πΊ))
4241biimpi 215 . . . . . . . 8 (𝑀 ∈ Word 𝑉 β†’ 𝑀 ∈ Word (Vtxβ€˜πΊ))
4342ad2antrl 726 . . . . . . 7 ((𝑋 ∈ 𝑉 ∧ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸))) β†’ 𝑀 ∈ Word (Vtxβ€˜πΊ))
44 fveq1 6890 . . . . . . . . . . . . . . 15 (𝑀 = βŸ¨β€œπ‘‹β€βŸ© β†’ (π‘€β€˜0) = (βŸ¨β€œπ‘‹β€βŸ©β€˜0))
45 s1fv 14590 . . . . . . . . . . . . . . 15 (𝑋 ∈ 𝑉 β†’ (βŸ¨β€œπ‘‹β€βŸ©β€˜0) = 𝑋)
4644, 45sylan9eq 2785 . . . . . . . . . . . . . 14 ((𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ 𝑋 ∈ 𝑉) β†’ (π‘€β€˜0) = 𝑋)
4746eqcomd 2731 . . . . . . . . . . . . 13 ((𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 = (π‘€β€˜0))
4847sneqd 4636 . . . . . . . . . . . 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 2819 . . . . . . . . . . 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 450 . . . . . . . . 9 ((𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸) β†’ (𝑋 ∈ 𝑉 β†’ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)))
5352adantl 480 . . . . . . . 8 ((𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)) β†’ (𝑋 ∈ 𝑉 β†’ {(π‘€β€˜0)} ∈ (Edgβ€˜πΊ)))
5453impcom 406 . . . . . . 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 411 . . . . . . . 8 (𝑀 = βŸ¨β€œπ‘‹β€βŸ© β†’ (𝑋 ∈ 𝑉 β†’ (π‘€β€˜0) = 𝑋))
5756ad2antrl 726 . . . . . . 7 ((𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)) β†’ (𝑋 ∈ 𝑉 β†’ (π‘€β€˜0) = 𝑋))
5857impcom 406 . . . . . 6 ((𝑋 ∈ 𝑉 ∧ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸))) β†’ (π‘€β€˜0) = 𝑋)
5955, 58jca 510 . . . . 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 799 . . . 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 282 . . 3 (𝑋 ∈ 𝑉 β†’ ((𝑀 ∈ (1 ClWWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑋) ↔ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸))))
6261rabbidva2 3421 . 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 2769 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3419  {csn 4624  β€˜cfv 6542  (class class class)co 7415  0cc0 11136  1c1 11137  β™―chash 14319  Word cword 14494  βŸ¨β€œcs1 14575  Vtxcvtx 28851  Edgcedg 28902   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-fz 13515  df-fzo 13658  df-hash 14320  df-word 14495  df-lsw 14543  df-s1 14576  df-clwwlk 29834  df-clwwlkn 29877  df-clwwlknon 29940
This theorem is referenced by:  clwwlknon1loop  29950  clwwlknon1nloop  29951
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