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Theorem iswwlkn 24662
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 24660 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( ( V WWalksN  E
) `  N )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) } )
21eleq2d 2513 . 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 5856 . . . 4  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
43eqeq1d 2445 . . 3  |-  ( w  =  W  ->  (
( # `  w )  =  ( N  + 
1 )  <->  ( # `  W
)  =  ( N  +  1 ) ) )
54elrab 3243 . 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 974    = wceq 1383    e. wcel 1804   {crab 2797   ` cfv 5578  (class class class)co 6281   1c1 9496    + caddc 9498   NN0cn0 10802   #chash 12387   WWalks cwwlk 24655   WWalksN cwwlkn 24656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-i2m1 9563  ax-1ne0 9564  ax-rrecex 9567  ax-cnre 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-nn 10544  df-n0 10803  df-wwlkn 24658
This theorem is referenced by:  wwlknimp  24665  wwlkn0  24667  wlklniswwlkn1  24677  wlklniswwlkn2  24678  wwlkiswwlkn  24680  vfwlkniswwlkn  24684  wwlknred  24701  wwlknext  24702  wwlkextproplem3  24721  clwwlkel  24771  clwwlkf  24772  wwlksubclwwlk  24782  rusgranumwlkl1  24925  rusgra0edg  24933
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