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Theorem clwwlkn 29788
Description: The set of closed walks of a fixed length 𝑁 as words over the set of vertices in a graph 𝐺. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2022.)
Assertion
Ref Expression
clwwlkn (𝑁 ClWWalksN 𝐺) = {𝑀 ∈ (ClWWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = 𝑁}
Distinct variable groups:   𝑀,𝐺   𝑀,𝑁

Proof of Theorem clwwlkn
Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6885 . . . . 5 (𝑔 = 𝐺 β†’ (ClWWalksβ€˜π‘”) = (ClWWalksβ€˜πΊ))
21adantl 481 . . . 4 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ (ClWWalksβ€˜π‘”) = (ClWWalksβ€˜πΊ))
3 eqeq2 2738 . . . . 5 (𝑛 = 𝑁 β†’ ((β™―β€˜π‘€) = 𝑛 ↔ (β™―β€˜π‘€) = 𝑁))
43adantr 480 . . . 4 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ ((β™―β€˜π‘€) = 𝑛 ↔ (β™―β€˜π‘€) = 𝑁))
52, 4rabeqbidv 3443 . . 3 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ {𝑀 ∈ (ClWWalksβ€˜π‘”) ∣ (β™―β€˜π‘€) = 𝑛} = {𝑀 ∈ (ClWWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = 𝑁})
6 df-clwwlkn 29787 . . 3 ClWWalksN = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ {𝑀 ∈ (ClWWalksβ€˜π‘”) ∣ (β™―β€˜π‘€) = 𝑛})
7 fvex 6898 . . . 4 (ClWWalksβ€˜πΊ) ∈ V
87rabex 5325 . . 3 {𝑀 ∈ (ClWWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = 𝑁} ∈ V
95, 6, 8ovmpoa 7559 . 2 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 ClWWalksN 𝐺) = {𝑀 ∈ (ClWWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = 𝑁})
106mpondm0 7644 . . 3 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 ClWWalksN 𝐺) = βˆ…)
11 eqid 2726 . . . . . . . . . . 11 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
1211clwwlkbp 29747 . . . . . . . . . 10 (𝑀 ∈ (ClWWalksβ€˜πΊ) β†’ (𝐺 ∈ V ∧ 𝑀 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑀 β‰  βˆ…))
1312simp2d 1140 . . . . . . . . 9 (𝑀 ∈ (ClWWalksβ€˜πΊ) β†’ 𝑀 ∈ Word (Vtxβ€˜πΊ))
14 lencl 14489 . . . . . . . . 9 (𝑀 ∈ Word (Vtxβ€˜πΊ) β†’ (β™―β€˜π‘€) ∈ β„•0)
1513, 14syl 17 . . . . . . . 8 (𝑀 ∈ (ClWWalksβ€˜πΊ) β†’ (β™―β€˜π‘€) ∈ β„•0)
16 eleq1 2815 . . . . . . . 8 ((β™―β€˜π‘€) = 𝑁 β†’ ((β™―β€˜π‘€) ∈ β„•0 ↔ 𝑁 ∈ β„•0))
1715, 16syl5ibcom 244 . . . . . . 7 (𝑀 ∈ (ClWWalksβ€˜πΊ) β†’ ((β™―β€˜π‘€) = 𝑁 β†’ 𝑁 ∈ β„•0))
1817con3rr3 155 . . . . . 6 (Β¬ 𝑁 ∈ β„•0 β†’ (𝑀 ∈ (ClWWalksβ€˜πΊ) β†’ Β¬ (β™―β€˜π‘€) = 𝑁))
1918ralrimiv 3139 . . . . 5 (Β¬ 𝑁 ∈ β„•0 β†’ βˆ€π‘€ ∈ (ClWWalksβ€˜πΊ) Β¬ (β™―β€˜π‘€) = 𝑁)
20 ral0 4507 . . . . . 6 βˆ€π‘€ ∈ βˆ… Β¬ (β™―β€˜π‘€) = 𝑁
21 fvprc 6877 . . . . . . 7 (Β¬ 𝐺 ∈ V β†’ (ClWWalksβ€˜πΊ) = βˆ…)
2221raleqdv 3319 . . . . . 6 (Β¬ 𝐺 ∈ V β†’ (βˆ€π‘€ ∈ (ClWWalksβ€˜πΊ) Β¬ (β™―β€˜π‘€) = 𝑁 ↔ βˆ€π‘€ ∈ βˆ… Β¬ (β™―β€˜π‘€) = 𝑁))
2320, 22mpbiri 258 . . . . 5 (Β¬ 𝐺 ∈ V β†’ βˆ€π‘€ ∈ (ClWWalksβ€˜πΊ) Β¬ (β™―β€˜π‘€) = 𝑁)
2419, 23jaoi 854 . . . 4 ((Β¬ 𝑁 ∈ β„•0 ∨ Β¬ 𝐺 ∈ V) β†’ βˆ€π‘€ ∈ (ClWWalksβ€˜πΊ) Β¬ (β™―β€˜π‘€) = 𝑁)
25 ianor 978 . . . 4 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ↔ (Β¬ 𝑁 ∈ β„•0 ∨ Β¬ 𝐺 ∈ V))
26 rabeq0 4379 . . . 4 ({𝑀 ∈ (ClWWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = 𝑁} = βˆ… ↔ βˆ€π‘€ ∈ (ClWWalksβ€˜πΊ) Β¬ (β™―β€˜π‘€) = 𝑁)
2724, 25, 263imtr4i 292 . . 3 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ {𝑀 ∈ (ClWWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = 𝑁} = βˆ…)
2810, 27eqtr4d 2769 . 2 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 ClWWalksN 𝐺) = {𝑀 ∈ (ClWWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = 𝑁})
299, 28pm2.61i 182 1 (𝑁 ClWWalksN 𝐺) = {𝑀 ∈ (ClWWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = 𝑁}
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055  {crab 3426  Vcvv 3468  βˆ…c0 4317  β€˜cfv 6537  (class class class)co 7405  β„•0cn0 12476  β™―chash 14295  Word cword 14470  Vtxcvtx 28764  ClWWalkscclwwlk 29743   ClWWalksN cclwwlkn 29786
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-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-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-hash 14296  df-word 14471  df-clwwlk 29744  df-clwwlkn 29787
This theorem is referenced by:  isclwwlkn  29789  clwwlkn0  29790  clwwlknfi  29807  clwlknf1oclwwlkn  29846
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