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Theorem clwwlkn 24558
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 998 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  ->  N  e.  NN0 )
2 ovex 6319 . . . 4  |-  ( V ClWWalks  E )  e.  _V
3 rabexg 4602 . . . 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 2482 . . . . 5  |-  ( n  =  N  ->  (
( # `  w )  =  n  <->  ( # `  w
)  =  N ) )
65rabbidv 3110 . . . 4  |-  ( n  =  N  ->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }  =  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  N } )
7 eqid 2467 . . . 4  |-  ( n  e.  NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
)  =  ( n  e.  NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
)
86, 7fvmptg 5954 . . 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 661 . 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 24543 . . . . . . 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 6303 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v ClWWalks  e )  =  ( V ClWWalks  E ) )
13 rabeq 3112 . . . . . . . . 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 4539 . . . . . . 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 466 . . . . . 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 3127 . . . . . . 7  |-  ( V  e.  X  ->  V  e.  _V )
1817adantr 465 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  V  e.  _V )
19 elex 3127 . . . . . . 7  |-  ( E  e.  Y  ->  E  e.  _V )
2019adantl 466 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  E  e.  _V )
21 nn0ex 10811 . . . . . . . 8  |-  NN0  e.  _V
2221mptex 6141 . . . . . . 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 6424 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V ClWWalksN  E )  =  ( n  e. 
NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
) )
2524fveq1d 5873 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V ClWWalksN  E ) `
 N )  =  ( ( n  e. 
NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
) `  N )
)
2625eqeq1d 2469 . . 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 1016 . 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 973    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118    |-> cmpt 4510   ` cfv 5593  (class class class)co 6294    |-> cmpt2 6296   NN0cn0 10805   #chash 12383   ClWWalks cclwwlk 24539   ClWWalksN cclwwlkn 24540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-i2m1 9570  ax-1ne0 9571  ax-rrecex 9574  ax-cnre 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-recs 7052  df-rdg 7086  df-nn 10547  df-n0 10806  df-clwwlkn 24543
This theorem is referenced by:  isclwwlkn  24560  clwwlknprop  24563  clwwlkn0  24565  clwwlknfi  24569
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