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Theorem wspthsn 27147
Description: The set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
Assertion
Ref Expression
wspthsn (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤}
Distinct variable groups:   𝑓,𝐺,𝑤   𝑤,𝑁
Allowed substitution hint:   𝑁(𝑓)

Proof of Theorem wspthsn
Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 6914 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑛 WWalksN 𝑔) = (𝑁 WWalksN 𝐺))
2 fveq2 6433 . . . . . . 7 (𝑔 = 𝐺 → (SPaths‘𝑔) = (SPaths‘𝐺))
32breqd 4884 . . . . . 6 (𝑔 = 𝐺 → (𝑓(SPaths‘𝑔)𝑤𝑓(SPaths‘𝐺)𝑤))
43exbidv 2020 . . . . 5 (𝑔 = 𝐺 → (∃𝑓 𝑓(SPaths‘𝑔)𝑤 ↔ ∃𝑓 𝑓(SPaths‘𝐺)𝑤))
54adantl 475 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → (∃𝑓 𝑓(SPaths‘𝑔)𝑤 ↔ ∃𝑓 𝑓(SPaths‘𝐺)𝑤))
61, 5rabeqbidv 3408 . . 3 ((𝑛 = 𝑁𝑔 = 𝐺) → {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
7 df-wspthsn 27132 . . 3 WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})
8 ovex 6937 . . . 4 (𝑁 WWalksN 𝐺) ∈ V
98rabex 5037 . . 3 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} ∈ V
106, 7, 9ovmpt2a 7051 . 2 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
117mpt2ndm0 7135 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = ∅)
12 df-wwlksn 27130 . . . . . 6 WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
1312mpt2ndm0 7135 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = ∅)
1413rabeqdv 3407 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
15 rab0 4185 . . . 4 {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = ∅
1614, 15syl6eq 2877 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = ∅)
1711, 16eqtr4d 2864 . 2 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
1810, 17pm2.61i 177 1 (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wa 386   = wceq 1656  wex 1878  wcel 2164  {crab 3121  Vcvv 3414  c0 4144   class class class wbr 4873  cfv 6123  (class class class)co 6905  1c1 10253   + caddc 10255  0cn0 11618  chash 13410  SPathscspths 27015  WWalkscwwlks 27124   WWalksN cwwlksn 27125   WSPathsN cwwspthsn 27127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-wwlksn 27130  df-wspthsn 27132
This theorem is referenced by:  iswspthn  27148  wspn0  27253
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