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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 6891 | . . . 4 β’ (π = πΊ β (Vtxβπ) = (VtxβπΊ)) | |
2 | eqidd 2728 | . . . 4 β’ (π = πΊ β β0 = β0) | |
3 | oveq2 7422 | . . . . 5 β’ (π = πΊ β (π ClWWalksN π) = (π ClWWalksN πΊ)) | |
4 | 3 | rabeqdv 3442 | . . . 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 7489 | . . 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 29885 | . . 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 6904 | . . . 4 β’ (VtxβπΊ) β V | |
8 | nn0ex 12500 | . . . 4 β’ β0 β V | |
9 | 7, 8 | mpoex 8078 | . . 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 6999 | . 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 6883 | . . 3 β’ (Β¬ πΊ β V β (ClWWalksNOnβπΊ) = β ) | |
12 | fvprc 6883 | . . . . 5 β’ (Β¬ πΊ β V β (VtxβπΊ) = β ) | |
13 | 12 | orcd 872 | . . . 4 β’ (Β¬ πΊ β V β ((VtxβπΊ) = β β¨ β0 = β )) |
14 | 0mpo0 7497 | . . . 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 2770 | . 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 846 = wceq 1534 β wcel 2099 {crab 3427 Vcvv 3469 β c0 4318 βcfv 6542 (class class class)co 7414 β cmpo 7416 0cc0 11130 β0cn0 12494 Vtxcvtx 28796 ClWWalksN cclwwlkn 29821 ClWWalksNOncclwwlknon 29884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-1cn 11188 ax-addcl 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 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-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-nn 12235 df-n0 12495 df-clwwlknon 29885 |
This theorem is referenced by: clwwlknon 29887 clwwlk0on0 29889 clwwlknon0 29890 2clwwlk2clwwlklem 30143 |
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