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Theorem clwwlkn 30442
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 990 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  ->  N  e.  NN0 )
2 ovex 6128 . . . 4  |-  ( V ClWWalks  E )  e.  _V
3 rabexg 4454 . . . 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 2452 . . . . 5  |-  ( n  =  N  ->  (
( # `  w )  =  n  <->  ( # `  w
)  =  N ) )
65rabbidv 2976 . . . 4  |-  ( n  =  N  ->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }  =  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  N } )
7 eqid 2443 . . . 4  |-  ( n  e.  NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
)  =  ( n  e.  NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
)
86, 7fvmptg 5784 . . 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 30429 . . . . . . 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 6112 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v ClWWalks  e )  =  ( V ClWWalks  E ) )
13 rabeq 2978 . . . . . . . . 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 4391 . . . . . . 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 2993 . . . . . . 7  |-  ( V  e.  X  ->  V  e.  _V )
1817adantr 465 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  V  e.  _V )
19 elex 2993 . . . . . . 7  |-  ( E  e.  Y  ->  E  e.  _V )
2019adantl 466 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  E  e.  _V )
21 nn0ex 10597 . . . . . . . 8  |-  NN0  e.  _V
2221mptex 5960 . . . . . . 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 6230 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V ClWWalksN  E )  =  ( n  e. 
NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
) )
2524fveq1d 5705 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V ClWWalksN  E ) `
 N )  =  ( ( n  e. 
NN0  |->  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  n }
) `  N )
)
2625eqeq1d 2451 . . 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 1008 . 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 965    = wceq 1369    e. wcel 1756   {crab 2731   _Vcvv 2984    e. cmpt 4362   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105   NN0cn0 10591   #chash 12115   ClWWalks cclwwlk 30425   ClWWalksN cclwwlkn 30426
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-i2m1 9362  ax-1ne0 9363  ax-rrecex 9366  ax-cnre 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-recs 6844  df-rdg 6878  df-nn 10335  df-n0 10592  df-clwwlkn 30429
This theorem is referenced by:  isclwwlkn  30444  clwwlknprop  30447  clwwlkn0  30449  clwwlknfi  30489
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