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Theorem numclwwlkovf 25223
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  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 5787 . . . 4  |-  ( n  =  N  ->  ( C `  n )  =  ( C `  N ) )
21adantl 464 . . 3  |-  ( ( v  =  X  /\  n  =  N )  ->  ( C `  n
)  =  ( C `
 N ) )
3 eqeq2 2407 . . . 4  |-  ( v  =  X  ->  (
( w `  0
)  =  v  <->  ( w `  0 )  =  X ) )
43adantr 463 . . 3  |-  ( ( v  =  X  /\  n  =  N )  ->  ( ( w ` 
0 )  =  v  <-> 
( w `  0
)  =  X ) )
52, 4rabeqbidv 3042 . 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 5797 . . 3  |-  ( C `
 N )  e. 
_V
87rabex 4529 . 2  |-  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X }  e.  _V
95, 6, 8ovmpt2a 6350 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 367    = wceq 1399    e. wcel 1836   {crab 2746    |-> cmpt 4438   ` cfv 5509  (class class class)co 6214    |-> cmpt2 6216   0cc0 9421   NN0cn0 10730   ClWWalksN cclwwlkn 24891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pr 4614
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-sbc 3266  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-sn 3958  df-pr 3960  df-op 3964  df-uni 4177  df-br 4381  df-opab 4439  df-id 4722  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-iota 5473  df-fun 5511  df-fv 5517  df-ov 6217  df-oprab 6218  df-mpt2 6219
This theorem is referenced by:  numclwwlkffin  25224  numclwwlkovfel2  25225  numclwwlkovf2  25226  extwwlkfab  25232  numclwwlkqhash  25242  numclwwlk3lem  25250  numclwwlk4  25252
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