Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  av-numclwwlkovf Structured version   Visualization version   GIF version

Theorem av-numclwwlkovf 41511
 Description: Value of operation 𝐹, mapping a vertex 𝑣 and a positive 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.) (Revised by AV, 28-May-2021.)
Hypothesis
Ref Expression
av-numclwwlkovf.f 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑣})
Assertion
Ref Expression
av-numclwwlkovf ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})
Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑤
Allowed substitution hints:   𝐹(𝑤,𝑣,𝑛)   𝑉(𝑤)

Proof of Theorem av-numclwwlkovf
StepHypRef Expression
1 oveq1 6556 . . . 4 (𝑛 = 𝑁 → (𝑛 ClWWalkSN 𝐺) = (𝑁 ClWWalkSN 𝐺))
21adantl 481 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → (𝑛 ClWWalkSN 𝐺) = (𝑁 ClWWalkSN 𝐺))
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 ∈ (𝑛 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑣} = {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})
6 av-numclwwlkovf.f . 2 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑣})
7 ovex 6577 . . 3 (𝑁 ClWWalkSN 𝐺) ∈ V
87rabex 4740 . 2 {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ V
95, 6, 8ovmpt2a 6689 1 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  ℕcn 10897   ClWWalkSN cclwwlksn 41184 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:  av-numclwwlkffin  41512  av-numclwwlkovfel2  41514  av-numclwwlkovf2  41515  av-extwwlkfab  41520  av-numclwwlkqhash  41530  av-numclwwlk3lem  41538  av-numclwwlk4  41540
 Copyright terms: Public domain W3C validator