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Theorem numclwwlkovf 30814
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 Huneke. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
Hypotheses
Ref Expression
numclwwlk.c  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
numclwwlk.f  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
Assertion
Ref Expression
numclwwlkovf  |-  ( ( X  e.  V  /\  N  e.  NN0 )  -> 
( X F N )  =  { w  e.  ( C `  N
)  |  ( 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 5791 . . . 4  |-  ( n  =  N  ->  ( C `  n )  =  ( C `  N ) )
21adantl 466 . . 3  |-  ( ( v  =  X  /\  n  =  N )  ->  ( C `  n
)  =  ( C `
 N ) )
3 eqeq2 2466 . . . 4  |-  ( v  =  X  ->  (
( w `  0
)  =  v  <->  ( w `  0 )  =  X ) )
43adantr 465 . . 3  |-  ( ( v  =  X  /\  n  =  N )  ->  ( ( w ` 
0 )  =  v  <-> 
( w `  0
)  =  X ) )
52, 4rabeqbidv 3065 . 2  |-  ( ( v  =  X  /\  n  =  N )  ->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v }  =  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X } )
6 numclwwlk.f . 2  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
7 fvex 5801 . . 3  |-  ( C `
 N )  e. 
_V
8 rabexg 4542 . . 3  |-  ( ( C `  N )  e.  _V  ->  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X }  e.  _V )
97, 8ax-mp 5 . 2  |-  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X }  e.  _V
105, 6, 9ovmpt2a 6323 1  |-  ( ( X  e.  V  /\  N  e.  NN0 )  -> 
( X F N )  =  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2799   _Vcvv 3070    |-> cmpt 4450   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   0cc0 9385   NN0cn0 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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