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Theorem iswspthsnon 29378
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 7449 . . 3 (π΄βˆ…π΅) = βˆ…
2 df-wspthsnon 29356 . . . . 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 7651 . . . 4 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 WSPathsNOn 𝐺) = βˆ…)
43oveqd 7429 . . 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 489 . . . . . 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 29376 . . . . . 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 3446 . . . 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 4382 . . . 4 {𝑀 ∈ βˆ… ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀} = βˆ…
1210, 11eqtrdi 2787 . . 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 2797 . 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 29374 . . . . . . . 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 480 . . . . . . 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 7429 . . . . . 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 2731 . . . . . . . 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 7651 . . . . . . 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 481 . . . . . 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 2771 . . . . 5 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) = βˆ…)
2120ex 412 . . . 4 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) = βˆ…))
224, 1eqtrdi 2787 . . . . 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 29377 . . . . . . . 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 407 . . . . . 6 ((Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡)) β†’ Β¬ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)
2827nexdv 1938 . . . . 5 ((Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝑀 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡)) β†’ Β¬ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)
2928ralrimiva 3145 . . . 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 4384 . . . 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 2774 . 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 480 . . 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 7421 . . . . 5 ((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) β†’ (π‘Ž(𝑁 WWalksNOn 𝐺)𝑏) = (𝐴(𝑁 WWalksNOn 𝐺)𝐡))
35 oveq12 7421 . . . . . . 7 ((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) β†’ (π‘Ž(SPathsOnβ€˜πΊ)𝑏) = (𝐴(SPathsOnβ€˜πΊ)𝐡))
3635breqd 5159 . . . . . 6 ((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) β†’ (𝑓(π‘Ž(SPathsOnβ€˜πΊ)𝑏)𝑀 ↔ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀))
3736exbidv 1923 . . . . 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 3448 . . . 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 481 . . 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 768 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ 𝐴 ∈ 𝑉)
41 simprr 770 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ 𝐡 ∈ 𝑉)
42 ovex 7445 . . . . 5 (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∈ V
4342rabex 5332 . . . 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 7563 . 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 1030 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 395   = wceq 1540  βˆƒwex 1780   ∈ wcel 2105  βˆ€wral 3060  {crab 3431  Vcvv 3473  βˆ…c0 4322   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412   ∈ cmpo 7414  β„•0cn0 12477  Vtxcvtx 28524  SPathsOncspthson 29240   WWalksNOn cwwlksnon 29349   WSPathsNOn cwwspthsnon 29351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-wwlksnon 29354  df-wspthsnon 29356
This theorem is referenced by:  wspthnon  29380  wpthswwlks2on  29483
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