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Theorem numclwwlkovg 30680
Description: Value of operation  G, mapping a vertex v and a nonnegative integer n to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v" 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 } )
numclwwlk.g  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
Assertion
Ref Expression
numclwwlkovg  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( X G N )  =  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) } )
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   
w, E    w, V    w, F
Allowed substitution hints:    E( v)    F( v, n)    G( w, v, n)

Proof of Theorem numclwwlkovg
StepHypRef Expression
1 fveq2 5691 . . . 4  |-  ( n  =  N  ->  ( C `  n )  =  ( C `  N ) )
21adantl 466 . . 3  |-  ( ( v  =  X  /\  n  =  N )  ->  ( C `  n
)  =  ( C `
 N ) )
3 eqeq2 2452 . . . 4  |-  ( v  =  X  ->  (
( w `  0
)  =  v  <->  ( w `  0 )  =  X ) )
4 oveq1 6098 . . . . . 6  |-  ( n  =  N  ->  (
n  -  2 )  =  ( N  - 
2 ) )
54fveq2d 5695 . . . . 5  |-  ( n  =  N  ->  (
w `  ( n  -  2 ) )  =  ( w `  ( N  -  2
) ) )
65eqeq1d 2451 . . . 4  |-  ( n  =  N  ->  (
( w `  (
n  -  2 ) )  =  ( w `
 0 )  <->  ( w `  ( N  -  2 ) )  =  ( w `  0 ) ) )
73, 6bi2anan9 868 . . 3  |-  ( ( v  =  X  /\  n  =  N )  ->  ( ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =  ( w `  0
) )  <->  ( (
w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) ) )
82, 7rabeqbidv 2967 . 2  |-  ( ( v  =  X  /\  n  =  N )  ->  { w  e.  ( C `  n )  |  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =  ( w `  0
) ) }  =  { w  e.  ( C `  N )  |  ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =  ( w `  0
) ) } )
9 numclwwlk.g . 2  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
10 fvex 5701 . . 3  |-  ( C `
 N )  e. 
_V
1110rabex 4443 . 2  |-  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) }  e.  _V
128, 9, 11ovmpt2a 6221 1  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( X G N )  =  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2719    e. cmpt 4350   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   0cc0 9282    - cmin 9595   2c2 10371   NN0cn0 10579   ZZ>=cuz 10861   ClWWalksN cclwwlkn 30414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096
This theorem is referenced by:  numclwwlkovgel  30681  extwwlkfab  30683  numclwwlk3lem  30701
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