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Theorem clwwlkn 30339
Description: The set of closed walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
Assertion
Ref Expression
clwwlkn  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( ( V ClWWalksN  E ) `
 N )  =  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  N } )
Distinct variable groups:    w, E    w, V    w, N
Allowed substitution hints:    X( w)    Y( w)

Proof of Theorem clwwlkn
Dummy variables  e  n  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 985 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  ->  N  e.  NN0 )
2 ovex 6115 . . . 4  |-  ( V ClWWalks  E )  e.  _V
3 rabexg 4439 . . . 4  |-  ( ( V ClWWalks  E )  e.  _V  ->  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  N }  e.  _V )
42, 3mp1i 12 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  ->  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  N }  e.  _V )
5 eqeq2 2450 . . . . 5  |-  ( n  =  N  ->  (
( # `  w )  =  n  <->  ( # `  w
)  =  N ) )
65rabbidv 2962 . . . 4  |-  ( n  =  N  ->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }  =  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  N } )
7 eqid 2441 . . . 4  |-  ( n  e.  NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
)  =  ( n  e.  NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
)
86, 7fvmptg 5769 . . 3  |-  ( ( N  e.  NN0  /\  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  N }  e.  _V )  ->  ( ( n  e.  NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
) `  N )  =  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  N } )
91, 4, 8syl2anc 656 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
) `  N )  =  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  N } )
10 df-clwwlkn 30326 . . . . . . 7  |- ClWWalksN  =  ( v  e.  _V , 
e  e.  _V  |->  ( n  e.  NN0  |->  { w  e.  ( v ClWWalks  e )  |  ( # `  w
)  =  n }
) )
1110a1i 11 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  -> ClWWalksN  =  ( v  e. 
_V ,  e  e. 
_V  |->  ( n  e. 
NN0  |->  { w  e.  ( v ClWWalks  e )  |  ( # `  w
)  =  n }
) ) )
12 oveq12 6099 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v ClWWalks  e )  =  ( V ClWWalks  E ) )
13 rabeq 2964 . . . . . . . . 9  |-  ( ( v ClWWalks  e )  =  ( V ClWWalks  E )  ->  { w  e.  ( v ClWWalks  e )  |  ( # `  w
)  =  n }  =  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  n } )
1412, 13syl 16 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  { w  e.  ( v ClWWalks  e )  |  ( # `  w
)  =  n }  =  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  n } )
1514mpteq2dv 4376 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( n  e.  NN0  |->  { w  e.  (
v ClWWalks  e )  |  (
# `  w )  =  n } )  =  ( n  e.  NN0  |->  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  n } ) )
1615adantl 463 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
( n  e.  NN0  |->  { w  e.  (
v ClWWalks  e )  |  (
# `  w )  =  n } )  =  ( n  e.  NN0  |->  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  n } ) )
17 elex 2979 . . . . . . 7  |-  ( V  e.  X  ->  V  e.  _V )
1817adantr 462 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  V  e.  _V )
19 elex 2979 . . . . . . 7  |-  ( E  e.  Y  ->  E  e.  _V )
2019adantl 463 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  E  e.  _V )
21 nn0ex 10581 . . . . . . . 8  |-  NN0  e.  _V
2221mptex 5945 . . . . . . 7  |-  ( n  e.  NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
)  e.  _V
2322a1i 11 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( n  e.  NN0  |->  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  n } )  e. 
_V )
2411, 16, 18, 20, 23ovmpt2d 6217 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V ClWWalksN  E )  =  ( n  e. 
NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
) )
2524fveq1d 5690 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V ClWWalksN  E ) `
 N )  =  ( ( n  e. 
NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
) `  N )
)
2625eqeq1d 2449 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( ( V ClWWalksN  E ) `  N
)  =  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  N }  <->  ( ( n  e.  NN0  |->  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  n } ) `  N )  =  {
w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  N } ) )
27263adant3 1003 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( ( ( V ClWWalksN  E ) `  N
)  =  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  N }  <->  ( ( n  e.  NN0  |->  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  n } ) `  N )  =  {
w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  N } ) )
289, 27mpbird 232 1  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( ( V ClWWalksN  E ) `
 N )  =  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  N } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   {crab 2717   _Vcvv 2970    e. cmpt 4347   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   NN0cn0 10575   #chash 12099   ClWWalks cclwwlk 30322   ClWWalksN cclwwlkn 30323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-i2m1 9346  ax-1ne0 9347  ax-rrecex 9350  ax-cnre 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-nn 10319  df-n0 10576  df-clwwlkn 30326
This theorem is referenced by:  isclwwlkn  30341  clwwlknprop  30344  clwwlkn0  30346  clwwlknfi  30386
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