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Theorem av-numclwwlkovf 41506
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 6534 . . . 4 (𝑛 = 𝑁 → (𝑛 ClWWalkSN 𝐺) = (𝑁 ClWWalkSN 𝐺))
21adantl 480 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → (𝑛 ClWWalkSN 𝐺) = (𝑁 ClWWalkSN 𝐺))
3 eqeq2 2620 . . . 4 (𝑣 = 𝑋 → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋))
43adantr 479 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋))
52, 4rabeqbidv 3167 . 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 6555 . . 3 (𝑁 ClWWalkSN 𝐺) ∈ V
87rabex 4735 . 2 {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ V
95, 6, 8ovmpt2a 6667 1 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  {crab 2899  cfv 5790  (class class class)co 6527  cmpt2 6529  0cc0 9792  cn 10867   ClWWalkSN cclwwlksn 41179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532
This theorem is referenced by:  av-numclwwlkffin  41507  av-numclwwlkovfel2  41509  av-numclwwlkovf2  41510  av-extwwlkfab  41515  av-numclwwlkqhash  41525  av-numclwwlk3lem  41533  av-numclwwlk4  41535
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