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Theorem numclwwlkfvc 30813
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 5794 . 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 5804 . 2  |-  ( ( V ClWWalksN  E ) `  N
)  e.  _V
41, 2, 3fvmpt 5878 1  |-  ( N  e.  NN0  ->  ( C `
 N )  =  ( ( V ClWWalksN  E ) `
 N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    |-> cmpt 4453   ` cfv 5521  (class class class)co 6195   NN0cn0 10685   ClWWalksN cclwwlkn 30557
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 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529
This theorem is referenced by:  extwwlkfablem2  30814  numclwwlkun  30815  numclwwlkffin  30818  numclwwlkovfel2  30819  numclwwlkovf2  30820  numclwwlkovf2ex  30822  numclwwlkovgel  30824  extwwlkfab  30826  numclwwlkqhash  30836  numclwwlk2lem1  30838  numclwlk2lem2f  30839  numclwlk2lem2f1o  30841  numclwwlk3lem  30844  numclwwlk4  30846  numclwwlk7  30850
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