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Theorem iswwlkn 24357
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 24355 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( ( V WWalksN  E
) `  N )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) } )
21eleq2d 2537 . 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 5864 . . . 4  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
43eqeq1d 2469 . . 3  |-  ( w  =  W  ->  (
( # `  w )  =  ( N  + 
1 )  <->  ( # `  W
)  =  ( N  +  1 ) ) )
54elrab 3261 . 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818   ` cfv 5586  (class class class)co 6282   1c1 9489    + caddc 9491   NN0cn0 10791   #chash 12367   WWalks cwwlk 24350   WWalksN cwwlkn 24351
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-i2m1 9556  ax-1ne0 9557  ax-rrecex 9560  ax-cnre 9561
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-nn 10533  df-n0 10792  df-wwlkn 24353
This theorem is referenced by:  wwlknimp  24360  wwlkn0  24362  wlklniswwlkn1  24372  wlklniswwlkn2  24373  wwlkiswwlkn  24375  vfwlkniswwlkn  24379  wwlknred  24396  wwlknext  24397  wwlkextproplem3  24416  clwwlkel  24466  clwwlkf  24467  wwlksubclwwlk  24477  rusgranumwlkl1  24620  rusgra0edg  24628
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