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Theorem wwlksn 26861
Description: The set of walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
Assertion
Ref Expression
wwlksn (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁

Proof of Theorem wwlksn
Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wwlksn 26855 . . . . 5 WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
21a1i 11 . . . 4 ((𝑁 ∈ ℕ0𝐺 ∈ V) → WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)}))
3 fveq2 6304 . . . . . . 7 (𝑔 = 𝐺 → (WWalks‘𝑔) = (WWalks‘𝐺))
43adantl 473 . . . . . 6 ((𝑛 = 𝑁𝑔 = 𝐺) → (WWalks‘𝑔) = (WWalks‘𝐺))
5 oveq1 6772 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
65eqeq2d 2734 . . . . . . 7 (𝑛 = 𝑁 → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1)))
76adantr 472 . . . . . 6 ((𝑛 = 𝑁𝑔 = 𝐺) → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1)))
84, 7rabeqbidv 3299 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)} = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
98adantl 473 . . . 4 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝑛 = 𝑁𝑔 = 𝐺)) → {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)} = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
10 simpl 474 . . . 4 ((𝑁 ∈ ℕ0𝐺 ∈ V) → 𝑁 ∈ ℕ0)
11 simpr 479 . . . 4 ((𝑁 ∈ ℕ0𝐺 ∈ V) → 𝐺 ∈ V)
12 fvex 6314 . . . . . 6 (WWalks‘𝐺) ∈ V
1312rabex 4920 . . . . 5 {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} ∈ V
1413a1i 11 . . . 4 ((𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} ∈ V)
152, 9, 10, 11, 14ovmpt2d 6905 . . 3 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
1615expcom 450 . 2 (𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}))
171reldmmpt2 6888 . . . . 5 Rel dom WWalksN
1817ovprc2 6800 . . . 4 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = ∅)
19 fvprc 6298 . . . . . 6 𝐺 ∈ V → (WWalks‘𝐺) = ∅)
2019rabeqdv 3298 . . . . 5 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)})
21 rab0 4063 . . . . 5 {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅
2220, 21syl6eq 2774 . . . 4 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅)
2318, 22eqtr4d 2761 . . 3 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
2423a1d 25 . 2 𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}))
2516, 24pm2.61i 176 1 (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  {crab 3018  Vcvv 3304  c0 4023  cfv 6001  (class class class)co 6765  cmpt2 6767  1c1 10050   + caddc 10052  0cn0 11405  chash 13232  WWalkscwwlks 26849   WWalksN cwwlksn 26850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-iota 5964  df-fun 6003  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-wwlksn 26855
This theorem is referenced by:  iswwlksn  26862  wwlksn0s  26891  0enwwlksnge1  26894  wwlksnfi  26945
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