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Theorem iswspthsnon 29099
Description: The set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)
Hypothesis
Ref Expression
iswspthsnon.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
iswspthsnon (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) = {𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀}
Distinct variable groups:   𝐴,𝑓,𝑀   𝐡,𝑓,𝑀   𝑓,𝐺,𝑀   𝑓,𝑁,𝑀   𝑓,𝑉,𝑀
Proof of Theorem iswspthsnon
Dummy variables π‘Ž 𝑏 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ov 7442 . . 3 (π΄βˆ…π΅) = βˆ…
2 df-wspthsnon 29077 . . . . 5 WSPathsNOn = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {𝑀 ∈ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀}))
32mpondm0 7643 . . . 4 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 WSPathsNOn 𝐺) = βˆ…)
43oveqd 7422 . . 3 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) = (π΄βˆ…π΅))
5 id 22 . . . . . . 7 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V))
65intnanrd 490 . . . . . 6 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ Β¬ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)))
7 iswspthsnon.v . . . . . . 7 𝑉 = (Vtxβ€˜πΊ)
87wwlksnon0 29097 . . . . . 6 (Β¬ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = βˆ…)
96, 8syl 17 . . . . 5 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = βˆ…)
109rabeqdv 3447 . . . 4 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ {𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀} = {𝑀 ∈ βˆ… ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀})
11 rab0 4381 . . . 4 {𝑀 ∈ βˆ… ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀} = βˆ…
1210, 11eqtrdi 2788 . . 3 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ {𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀} = βˆ…)
131, 4, 123eqtr4a 2798 . 2 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) = {𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀})
147wspthsnon 29095 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 WSPathsNOn 𝐺) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀}))
1514adantr 481 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝑁 WSPathsNOn 𝐺) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀}))
1615oveqd 7422 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) = (𝐴(π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀})𝐡))
17 eqid 2732 . . . . . . . 8 (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀}) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀})
1817mpondm0 7643 . . . . . . 7 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀})𝐡) = βˆ…)
1918adantl 482 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀})𝐡) = βˆ…)
2016, 19eqtrd 2772 . . . . 5 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) = βˆ…)
2120ex 413 . . . 4 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) = βˆ…))
224, 1eqtrdi 2788 . . . . 5 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) = βˆ…)
2322a1d 25 . . . 4 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) = βˆ…))
2421, 23pm2.61i 182 . . 3 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) = βˆ…)
257wwlksonvtx 29098 . . . . . . . 8 (𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) β†’ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉))
2625pm2.24d 151 . . . . . . 7 (𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) β†’ (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ Β¬ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀))
2726impcom 408 . . . . . 6 ((Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡)) β†’ Β¬ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)
2827nexdv 1939 . . . . 5 ((Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡)) β†’ Β¬ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)
2928ralrimiva 3146 . . . 4 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ βˆ€π‘€ ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) Β¬ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)
30 rabeq0 4383 . . . 4 ({𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀} = βˆ… ↔ βˆ€π‘€ ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) Β¬ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)
3129, 30sylibr 233 . . 3 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ {𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀} = βˆ…)
3224, 31eqtr4d 2775 . 2 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) = {𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀})
3314adantr 481 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝑁 WSPathsNOn 𝐺) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀}))
34 oveq12 7414 . . . . 5 ((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) β†’ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) = (𝐴(𝑁 WWalksNOn 𝐺)𝐡))
35 oveq12 7414 . . . . . . 7 ((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) β†’ (π‘Ž(SPathsOnβ€˜πΊ)𝑏) = (𝐴(SPathsOnβ€˜πΊ)𝐡))
3635breqd 5158 . . . . . 6 ((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) β†’ (𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀 ↔ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀))
3736exbidv 1924 . . . . 5 ((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) β†’ (βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀 ↔ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀))
3834, 37rabeqbidv 3449 . . . 4 ((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) β†’ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀} = {𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀})
3938adantl 482 . . 3 ((((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) ∧ (π‘Ž = 𝐴 ∧ 𝑏 = 𝐡)) β†’ {𝑀 ∈ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀} = {𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀})
40 simprl 769 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ 𝐴 ∈ 𝑉)
41 simprr 771 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ 𝐡 ∈ 𝑉)
42 ovex 7438 . . . . 5 (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∈ V
4342rabex 5331 . . . 4 {𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀} ∈ V
4443a1i 11 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ {𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀} ∈ V)
4533, 39, 40, 41, 44ovmpod 7556 . 2 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) = {𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀})
4613, 32, 45ecase 1031 1 (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) = {𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀}
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  βˆ€wral 3061  {crab 3432  Vcvv 3474  βˆ…c0 4321   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  β„•0cn0 12468  Vtxcvtx 28245  SPathsOncspthson 28961   WWalksNOn cwwlksnon 29070   WSPathsNOn cwwspthsnon 29072
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  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-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-wwlksnon 29075  df-wspthsnon 29077
This theorem is referenced by:  wspthnon  29101  wpthswwlks2on  29204
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