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Theorem numclwwlkfvc 25198
Description: Value of function  C, mapping a nonnegative number n to the closed walks having length n. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
Hypothesis
Ref Expression
numclwwlk.c  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
Assertion
Ref Expression
numclwwlkfvc  |-  ( N  e.  NN0  ->  ( C `
 N )  =  ( ( V ClWWalksN  E ) `
 N ) )
Distinct variable groups:    n, E    n, N    n, V
Allowed substitution hint:    C( n)

Proof of Theorem numclwwlkfvc
StepHypRef Expression
1 fveq2 5774 . 2  |-  ( n  =  N  ->  (
( V ClWWalksN  E ) `  n )  =  ( ( V ClWWalksN  E ) `  N ) )
2 numclwwlk.c . 2  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
3 fvex 5784 . 2  |-  ( ( V ClWWalksN  E ) `  N
)  e.  _V
41, 2, 3fvmpt 5857 1  |-  ( N  e.  NN0  ->  ( C `
 N )  =  ( ( V ClWWalksN  E ) `
 N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826    |-> cmpt 4425   ` cfv 5496  (class class class)co 6196   NN0cn0 10712   ClWWalksN cclwwlkn 24870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504
This theorem is referenced by:  extwwlkfablem2  25199  numclwwlkun  25200  numclwwlkffin  25203  numclwwlkovfel2  25204  numclwwlkovf2  25205  numclwwlkovf2ex  25207  numclwwlkovgel  25209  extwwlkfab  25211  numclwwlkqhash  25221  numclwwlk2lem1  25223  numclwlk2lem2f  25224  numclwlk2lem2f1o  25226  numclwwlk3lem  25229  numclwwlk4  25231  numclwwlk7  25235
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