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Mirrors > Home > MPE Home > Th. List > clwwlknonmpo | Structured version Visualization version GIF version |
Description: (ClWWalksNOnβπΊ) is an operator mapping a vertex π£ and a nonnegative integer π to the set of closed walks on π£ of length π as words over the set of vertices in a graph πΊ. (Contributed by AV, 25-Feb-2022.) (Proof shortened by AV, 2-Mar-2024.) |
Ref | Expression |
---|---|
clwwlknonmpo | β’ (ClWWalksNOnβπΊ) = (π£ β (VtxβπΊ), π β β0 β¦ {π€ β (π ClWWalksN πΊ) β£ (π€β0) = π£}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6839 | . . . 4 β’ (π = πΊ β (Vtxβπ) = (VtxβπΊ)) | |
2 | eqidd 2738 | . . . 4 β’ (π = πΊ β β0 = β0) | |
3 | oveq2 7359 | . . . . 5 β’ (π = πΊ β (π ClWWalksN π) = (π ClWWalksN πΊ)) | |
4 | 3 | rabeqdv 3420 | . . . 4 β’ (π = πΊ β {π€ β (π ClWWalksN π) β£ (π€β0) = π£} = {π€ β (π ClWWalksN πΊ) β£ (π€β0) = π£}) |
5 | 1, 2, 4 | mpoeq123dv 7426 | . . 3 β’ (π = πΊ β (π£ β (Vtxβπ), π β β0 β¦ {π€ β (π ClWWalksN π) β£ (π€β0) = π£}) = (π£ β (VtxβπΊ), π β β0 β¦ {π€ β (π ClWWalksN πΊ) β£ (π€β0) = π£})) |
6 | df-clwwlknon 28861 | . . 3 β’ ClWWalksNOn = (π β V β¦ (π£ β (Vtxβπ), π β β0 β¦ {π€ β (π ClWWalksN π) β£ (π€β0) = π£})) | |
7 | fvex 6852 | . . . 4 β’ (VtxβπΊ) β V | |
8 | nn0ex 12377 | . . . 4 β’ β0 β V | |
9 | 7, 8 | mpoex 8004 | . . 3 β’ (π£ β (VtxβπΊ), π β β0 β¦ {π€ β (π ClWWalksN πΊ) β£ (π€β0) = π£}) β V |
10 | 5, 6, 9 | fvmpt 6945 | . 2 β’ (πΊ β V β (ClWWalksNOnβπΊ) = (π£ β (VtxβπΊ), π β β0 β¦ {π€ β (π ClWWalksN πΊ) β£ (π€β0) = π£})) |
11 | fvprc 6831 | . . 3 β’ (Β¬ πΊ β V β (ClWWalksNOnβπΊ) = β ) | |
12 | fvprc 6831 | . . . . 5 β’ (Β¬ πΊ β V β (VtxβπΊ) = β ) | |
13 | 12 | orcd 871 | . . . 4 β’ (Β¬ πΊ β V β ((VtxβπΊ) = β β¨ β0 = β )) |
14 | 0mpo0 7434 | . . . 4 β’ (((VtxβπΊ) = β β¨ β0 = β ) β (π£ β (VtxβπΊ), π β β0 β¦ {π€ β (π ClWWalksN πΊ) β£ (π€β0) = π£}) = β ) | |
15 | 13, 14 | syl 17 | . . 3 β’ (Β¬ πΊ β V β (π£ β (VtxβπΊ), π β β0 β¦ {π€ β (π ClWWalksN πΊ) β£ (π€β0) = π£}) = β ) |
16 | 11, 15 | eqtr4d 2780 | . 2 β’ (Β¬ πΊ β V β (ClWWalksNOnβπΊ) = (π£ β (VtxβπΊ), π β β0 β¦ {π€ β (π ClWWalksN πΊ) β£ (π€β0) = π£})) |
17 | 10, 16 | pm2.61i 182 | 1 β’ (ClWWalksNOnβπΊ) = (π£ β (VtxβπΊ), π β β0 β¦ {π€ β (π ClWWalksN πΊ) β£ (π€β0) = π£}) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β¨ wo 845 = wceq 1541 β wcel 2106 {crab 3405 Vcvv 3443 β c0 4280 βcfv 6493 (class class class)co 7351 β cmpo 7353 0cc0 11009 β0cn0 12371 Vtxcvtx 27776 ClWWalksN cclwwlkn 28797 ClWWalksNOncclwwlknon 28860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-1cn 11067 ax-addcl 11069 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-nn 12112 df-n0 12372 df-clwwlknon 28861 |
This theorem is referenced by: clwwlknon 28863 clwwlk0on0 28865 clwwlknon0 28866 2clwwlk2clwwlklem 29119 |
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