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Theorem isclwwlkn 25342
Description: Properties of a word to represent a closed walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.)
Assertion
Ref Expression
isclwwlkn  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( W  e.  ( ( V ClWWalksN  E ) `  N )  <->  ( W  e.  ( V ClWWalks  E )  /\  ( # `  W
)  =  N ) ) )

Proof of Theorem isclwwlkn
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 clwwlkn 25340 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( ( V ClWWalksN  E ) `
 N )  =  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  N } )
21eleq2d 2499 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( W  e.  ( ( V ClWWalksN  E ) `  N )  <->  W  e.  { w  e.  ( V ClWWalks  E )  |  (
# `  w )  =  N } ) )
3 fveq2 5881 . . . 4  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
43eqeq1d 2431 . . 3  |-  ( w  =  W  ->  (
( # `  w )  =  N  <->  ( # `  W
)  =  N ) )
54elrab 3235 . 2  |-  ( W  e.  { w  e.  ( V ClWWalks  E )  |  ( # `  w
)  =  N }  <->  ( W  e.  ( V ClWWalks  E )  /\  ( # `
 W )  =  N ) )
62, 5syl6bb 264 1  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( W  e.  ( ( V ClWWalksN  E ) `  N )  <->  ( W  e.  ( V ClWWalks  E )  /\  ( # `  W
)  =  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   {crab 2786   ` cfv 5601  (class class class)co 6305   NN0cn0 10869   #chash 12512   ClWWalks cclwwlk 25321   ClWWalksN cclwwlkn 25322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-i2m1 9606  ax-1ne0 9607  ax-rrecex 9610  ax-cnre 9611
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-nn 10610  df-n0 10870  df-clwwlkn 25325
This theorem is referenced by:  clwwlkn2  25348  clwwlknimp  25349  clwwlkisclwwlkn  25364  clwwlkf  25367  clwwlkext2edg  25375  wwlkext2clwwlk  25376  clwwnisshclwwn  25382  clwlkfclwwlk  25417  clwlkfoclwwlk  25418  extwwlkfablem2  25651  numclwwlkovfel2  25656  numclwwlkovf2ex  25659  numclwwlkovgelim  25662
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