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Theorem clwwlknclwwlkdif 28388
Description: The set 𝐴 of walks of length 𝑁 starting with a fixed vertex 𝑉 and ending not at this vertex is the difference between the set 𝐢 of walks of length 𝑁 starting with this vertex 𝑋 and the set 𝐡 of closed walks of length 𝑁 anchored at this vertex 𝑋. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.) (Revised by AV, 16-Mar-2022.)
Hypotheses
Ref Expression
clwwlknclwwlkdif.a 𝐴 = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)}
clwwlknclwwlkdif.b 𝐡 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)
clwwlknclwwlkdif.c 𝐢 = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}
Assertion
Ref Expression
clwwlknclwwlkdif 𝐴 = (𝐢 βˆ– 𝐡)
Distinct variable groups:   𝑀,𝐺   𝑀,𝑁   𝑀,𝑋
Allowed substitution hints:   𝐴(𝑀)   𝐡(𝑀)   𝐢(𝑀)

Proof of Theorem clwwlknclwwlkdif
StepHypRef Expression
1 clwwlknclwwlkdif.a . 2 𝐴 = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)}
2 clwwlknclwwlkdif.c . . . 4 𝐢 = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}
3 clwwlknclwwlkdif.b . . . . 5 𝐡 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)
4 eqid 2736 . . . . . 6 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
54iswwlksnon 28263 . . . . 5 (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋)}
63, 5eqtri 2764 . . . 4 𝐡 = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋)}
72, 6difeq12i 4061 . . 3 (𝐢 βˆ– 𝐡) = ({𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} βˆ– {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋)})
8 difrab 4248 . . 3 ({𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} βˆ– {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋)}) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ Β¬ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋))}
9 annotanannot 833 . . . . 5 (((π‘€β€˜0) = 𝑋 ∧ Β¬ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋)) ↔ ((π‘€β€˜0) = 𝑋 ∧ Β¬ (π‘€β€˜π‘) = 𝑋))
10 df-ne 2942 . . . . . . 7 ((π‘€β€˜π‘) β‰  𝑋 ↔ Β¬ (π‘€β€˜π‘) = 𝑋)
11 wwlknlsw 28257 . . . . . . . 8 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ (π‘€β€˜π‘) = (lastSβ€˜π‘€))
1211neeq1d 3001 . . . . . . 7 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ ((π‘€β€˜π‘) β‰  𝑋 ↔ (lastSβ€˜π‘€) β‰  𝑋))
1310, 12bitr3id 285 . . . . . 6 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ (Β¬ (π‘€β€˜π‘) = 𝑋 ↔ (lastSβ€˜π‘€) β‰  𝑋))
1413anbi2d 630 . . . . 5 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ (((π‘€β€˜0) = 𝑋 ∧ Β¬ (π‘€β€˜π‘) = 𝑋) ↔ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)))
159, 14bitrid 283 . . . 4 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ (((π‘€β€˜0) = 𝑋 ∧ Β¬ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋)) ↔ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)))
1615rabbiia 3414 . . 3 {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ Β¬ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜π‘) = 𝑋))} = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)}
177, 8, 163eqtrri 2769 . 2 {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)} = (𝐢 βˆ– 𝐡)
181, 17eqtri 2764 1 𝐴 = (𝐢 βˆ– 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 397   = wceq 1539   ∈ wcel 2104   β‰  wne 2941  {crab 3284   βˆ– cdif 3889  β€˜cfv 6458  (class class class)co 7307  0cc0 10917  lastSclsw 14310  Vtxcvtx 27411   WWalksN cwwlksn 28236   WWalksNOn cwwlksnon 28237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-cnex 10973  ax-resscn 10974  ax-1cn 10975  ax-icn 10976  ax-addcl 10977  ax-addrcl 10978  ax-mulcl 10979  ax-mulrcl 10980  ax-mulcom 10981  ax-addass 10982  ax-mulass 10983  ax-distr 10984  ax-i2m1 10985  ax-1ne0 10986  ax-1rid 10987  ax-rnegex 10988  ax-rrecex 10989  ax-cnre 10990  ax-pre-lttri 10991  ax-pre-lttrn 10992  ax-pre-ltadd 10993  ax-pre-mulgt0 10994
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-1st 7863  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-er 8529  df-map 8648  df-en 8765  df-dom 8766  df-sdom 8767  df-fin 8768  df-card 9741  df-pnf 11057  df-mnf 11058  df-xr 11059  df-ltxr 11060  df-le 11061  df-sub 11253  df-neg 11254  df-nn 12020  df-n0 12280  df-z 12366  df-uz 12629  df-fz 13286  df-fzo 13429  df-hash 14091  df-word 14263  df-lsw 14311  df-wwlks 28240  df-wwlksn 28241  df-wwlksnon 28242
This theorem is referenced by:  clwwlknclwwlkdifnum  28389
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