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Theorem iswwlkn 30467
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 30465 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( ( V WWalksN  E
) `  N )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) } )
21eleq2d 2524 . 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 5800 . . . 4  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
43eqeq1d 2456 . . 3  |-  ( w  =  W  ->  (
( # `  w )  =  ( N  + 
1 )  <->  ( # `  W
)  =  ( N  +  1 ) ) )
54elrab 3224 . 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 965    = wceq 1370    e. wcel 1758   {crab 2803   ` cfv 5527  (class class class)co 6201   1c1 9395    + caddc 9397   NN0cn0 10691   #chash 12221   WWalks cwwlk 30460   WWalksN cwwlkn 30461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-i2m1 9462  ax-1ne0 9463  ax-rrecex 9466  ax-cnre 9467
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-recs 6943  df-rdg 6977  df-nn 10435  df-n0 10692  df-wwlkn 30463
This theorem is referenced by:  wwlknimp  30470  wwlkn0  30472  wlklniswwlkn1  30482  wlklniswwlkn2  30483  wwlkiswwlkn  30485  vfwlkniswwlkn  30489  wwlknred  30504  wwlknext  30505  clwwlkel  30604  clwwlkf  30605  wwlksubclwwlk  30615  rusgranumwlkl1  30708  wwlkextproplem3  30711  rusgra0edg  30722
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