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Theorem numclwwlkovf 30814
 Description: Value of operation , 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 Huneke. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
Hypotheses
Ref Expression
numclwwlk.c ClWWalksN
numclwwlk.f
Assertion
Ref Expression
numclwwlkovf
Distinct variable groups:   ,   ,   ,   ,   ,   ,,,   ,   ,,,   ,
Allowed substitution hints:   (,)   (,,)   ()

Proof of Theorem numclwwlkovf
StepHypRef Expression
1 fveq2 5791 . . . 4
21adantl 466 . . 3
3 eqeq2 2466 . . . 4
43adantr 465 . . 3
52, 4rabeqbidv 3065 . 2
6 numclwwlk.f . 2
7 fvex 5801 . . 3
8 rabexg 4542 . . 3
97, 8ax-mp 5 . 2
105, 6, 9ovmpt2a 6323 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1370   wcel 1758  crab 2799  cvv 3070   cmpt 4450  cfv 5518  (class class class)co 6192   cmpt2 6194  cc0 9385  cn0 10682   ClWWalksN cclwwlkn 30554 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197 This theorem is referenced by:  numclwwlkffin  30815  numclwwlkovfel2  30816  numclwwlkovf2  30817  extwwlkfab  30823  numclwwlkqhash  30833  numclwwlk3lem  30841  numclwwlk4  30843
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