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Theorem iswwlksnon 29374
Description: The set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)
Hypothesis
Ref Expression
iswwlksnon.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
iswwlksnon (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)}
Distinct variable groups:   𝑀,𝐴   𝑀,𝐡   𝑀,𝐺   𝑀,𝑁   𝑀,𝑉

Proof of Theorem iswwlksnon
Dummy variables π‘Ž 𝑏 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ov 7448 . . 3 (π΄βˆ…π΅) = βˆ…
2 df-wwlksnon 29353 . . . . 5 WWalksNOn = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {𝑀 ∈ (𝑛 WWalksN 𝑔) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘›) = 𝑏)}))
32mpondm0 7649 . . . 4 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 WWalksNOn 𝐺) = βˆ…)
43oveqd 7428 . . 3 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = (π΄βˆ…π΅))
5 df-wwlksn 29352 . . . . . 6 WWalksN = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ {𝑀 ∈ (WWalksβ€˜π‘”) ∣ (β™―β€˜π‘€) = (𝑛 + 1)})
65mpondm0 7649 . . . . 5 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 WWalksN 𝐺) = βˆ…)
76rabeqdv 3445 . . . 4 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} = {𝑀 ∈ βˆ… ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)})
8 rab0 4381 . . . 4 {𝑀 ∈ βˆ… ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} = βˆ…
97, 8eqtrdi 2786 . . 3 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} = βˆ…)
101, 4, 93eqtr4a 2796 . 2 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)})
11 iswwlksnon.v . . . . . . . . 9 𝑉 = (Vtxβ€˜πΊ)
1211wwlksnon 29372 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 WWalksNOn 𝐺) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)}))
1312adantr 479 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝑁 WWalksNOn 𝐺) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)}))
1413oveqd 7428 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = (𝐴(π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)})𝐡))
15 eqid 2730 . . . . . . . 8 (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)}) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)})
1615mpondm0 7649 . . . . . . 7 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)})𝐡) = βˆ…)
1716adantl 480 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)})𝐡) = βˆ…)
1814, 17eqtrd 2770 . . . . 5 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = βˆ…)
1918ex 411 . . . 4 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = βˆ…))
204, 1eqtrdi 2786 . . . . 5 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = βˆ…)
2120a1d 25 . . . 4 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = βˆ…))
2219, 21pm2.61i 182 . . 3 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = βˆ…)
2311wwlknllvtx 29367 . . . . . . 7 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ ((π‘€β€˜0) ∈ 𝑉 ∧ (π‘€β€˜π‘) ∈ 𝑉))
24 eleq1 2819 . . . . . . . . 9 (𝐴 = (π‘€β€˜0) β†’ (𝐴 ∈ 𝑉 ↔ (π‘€β€˜0) ∈ 𝑉))
2524eqcoms 2738 . . . . . . . 8 ((π‘€β€˜0) = 𝐴 β†’ (𝐴 ∈ 𝑉 ↔ (π‘€β€˜0) ∈ 𝑉))
26 eleq1 2819 . . . . . . . . 9 (𝐡 = (π‘€β€˜π‘) β†’ (𝐡 ∈ 𝑉 ↔ (π‘€β€˜π‘) ∈ 𝑉))
2726eqcoms 2738 . . . . . . . 8 ((π‘€β€˜π‘) = 𝐡 β†’ (𝐡 ∈ 𝑉 ↔ (π‘€β€˜π‘) ∈ 𝑉))
2825, 27bi2anan9 635 . . . . . . 7 (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡) β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ↔ ((π‘€β€˜0) ∈ 𝑉 ∧ (π‘€β€˜π‘) ∈ 𝑉)))
2923, 28syl5ibrcom 246 . . . . . 6 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡) β†’ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)))
3029con3rr3 155 . . . . 5 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ Β¬ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)))
3130ralrimiv 3143 . . . 4 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ βˆ€π‘€ ∈ (𝑁 WWalksN 𝐺) Β¬ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡))
32 rabeq0 4383 . . . 4 ({𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} = βˆ… ↔ βˆ€π‘€ ∈ (𝑁 WWalksN 𝐺) Β¬ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡))
3331, 32sylibr 233 . . 3 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} = βˆ…)
3422, 33eqtr4d 2773 . 2 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)})
3512adantr 479 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝑁 WWalksNOn 𝐺) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)}))
36 eqeq2 2742 . . . . . 6 (π‘Ž = 𝐴 β†’ ((π‘€β€˜0) = π‘Ž ↔ (π‘€β€˜0) = 𝐴))
37 eqeq2 2742 . . . . . 6 (𝑏 = 𝐡 β†’ ((π‘€β€˜π‘) = 𝑏 ↔ (π‘€β€˜π‘) = 𝐡))
3836, 37bi2anan9 635 . . . . 5 ((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) β†’ (((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏) ↔ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)))
3938rabbidv 3438 . . . 4 ((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)} = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)})
4039adantl 480 . . 3 ((((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) ∧ (π‘Ž = 𝐴 ∧ 𝑏 = 𝐡)) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)} = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)})
41 simprl 767 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ 𝐴 ∈ 𝑉)
42 simprr 769 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ 𝐡 ∈ 𝑉)
43 ovex 7444 . . . . 5 (𝑁 WWalksN 𝐺) ∈ V
4443rabex 5331 . . . 4 {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} ∈ V
4544a1i 11 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} ∈ V)
4635, 40, 41, 42, 45ovmpod 7562 . 2 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)})
4710, 34, 46ecase 1029 1 (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)}
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  {crab 3430  Vcvv 3472  βˆ…c0 4321  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  0cc0 11112  1c1 11113   + caddc 11115  β„•0cn0 12476  β™―chash 14294  Vtxcvtx 28523  WWalkscwwlks 29346   WWalksN cwwlksn 29347   WWalksNOn cwwlksnon 29348
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 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  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 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  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 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-wwlks 29351  df-wwlksn 29352  df-wwlksnon 29353
This theorem is referenced by:  wwlknon  29378  wwlksnonfi  29441  wpthswwlks2on  29482  clwwlknclwwlkdif  29499  clwwlknclwwlkdifnum  29500  numclwwlkqhash  29895
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