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Theorem iswwlkn 24982
Description: Properties of a word to represent a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Assertion
Ref Expression
iswwlkn  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( W  e.  ( ( V WWalksN  E ) `  N )  <->  ( W  e.  ( V WWalks  E )  /\  ( # `  W
)  =  ( N  +  1 ) ) ) )

Proof of Theorem iswwlkn
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 wwlkn 24980 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( ( V WWalksN  E
) `  N )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) } )
21eleq2d 2472 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( W  e.  ( ( V WWalksN  E ) `  N )  <->  W  e.  { w  e.  ( V WWalks  E )  |  (
# `  w )  =  ( N  + 
1 ) } ) )
3 fveq2 5805 . . . 4  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
43eqeq1d 2404 . . 3  |-  ( w  =  W  ->  (
( # `  w )  =  ( N  + 
1 )  <->  ( # `  W
)  =  ( N  +  1 ) ) )
54elrab 3206 . 2  |-  ( W  e.  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) }  <-> 
( W  e.  ( V WWalks  E )  /\  ( # `  W )  =  ( N  + 
1 ) ) )
62, 5syl6bb 261 1  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( W  e.  ( ( V WWalksN  E ) `  N )  <->  ( W  e.  ( V WWalks  E )  /\  ( # `  W
)  =  ( N  +  1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   {crab 2757   ` cfv 5525  (class class class)co 6234   1c1 9443    + caddc 9445   NN0cn0 10756   #chash 12359   WWalks cwwlk 24975   WWalksN cwwlkn 24976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-i2m1 9510  ax-1ne0 9511  ax-rrecex 9514  ax-cnre 9515
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-recs 6999  df-rdg 7033  df-nn 10497  df-n0 10757  df-wwlkn 24978
This theorem is referenced by:  wwlknimp  24985  wwlkn0  24987  wlklniswwlkn1  24997  wlklniswwlkn2  24998  wwlkiswwlkn  25000  vfwlkniswwlkn  25004  wwlknred  25021  wwlknext  25022  wwlkextproplem3  25041  clwwlkel  25091  clwwlkf  25092  wwlksubclwwlk  25102  rusgranumwlkl1  25245  rusgra0edg  25253
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