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Theorem numclwwlkovf 26608
 Description: Value of operation 𝐹, mapping a vertex v and a nonnegative integer n to the "(For a fixed vertex v, let f(n) be the number of) walks from v to v of length n" according to definition 5 in [Huneke] p. 2. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
Hypotheses
Ref Expression
numclwwlk.c 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
numclwwlk.f 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})
Assertion
Ref Expression
numclwwlkovf ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑋})
Distinct variable groups:   𝑛,𝐸   𝑛,𝑁   𝑛,𝑉   𝑤,𝐶   𝑤,𝑁   𝐶,𝑛,𝑣,𝑤   𝑣,𝑁   𝑛,𝑋,𝑣,𝑤   𝑣,𝑉
Allowed substitution hints:   𝐸(𝑤,𝑣)   𝐹(𝑤,𝑣,𝑛)   𝑉(𝑤)

Proof of Theorem numclwwlkovf
StepHypRef Expression
1 fveq2 6103 . . . 4 (𝑛 = 𝑁 → (𝐶𝑛) = (𝐶𝑁))
21adantl 481 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → (𝐶𝑛) = (𝐶𝑁))
3 eqeq2 2621 . . . 4 (𝑣 = 𝑋 → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋))
43adantr 480 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋))
52, 4rabeqbidv 3168 . 2 ((𝑣 = 𝑋𝑛 = 𝑁) → {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣} = {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑋})
6 numclwwlk.f . 2 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})
7 fvex 6113 . . 3 (𝐶𝑁) ∈ V
87rabex 4740 . 2 {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑋} ∈ V
95, 6, 8ovmpt2a 6689 1 ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑋})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  ℕ0cn0 11169   ClWWalksN cclwwlkn 26277 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  numclwwlkffin  26609  numclwwlkovfel2  26610  numclwwlkovf2  26611  extwwlkfab  26617  numclwwlkqhash  26627  numclwwlk3lem  26635  numclwwlk4  26637
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