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Theorem iswwlkn 25412
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 25410 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( ( V WWalksN  E
) `  N )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) } )
21eleq2d 2514 . 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 5865 . . . 4  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
43eqeq1d 2453 . . 3  |-  ( w  =  W  ->  (
( # `  w )  =  ( N  + 
1 )  <->  ( # `  W
)  =  ( N  +  1 ) ) )
54elrab 3196 . 2  |-  ( W  e.  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) }  <-> 
( W  e.  ( V WWalks  E )  /\  ( # `  W )  =  ( N  + 
1 ) ) )
62, 5syl6bb 265 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   {crab 2741   ` cfv 5582  (class class class)co 6290   1c1 9540    + caddc 9542   NN0cn0 10869   #chash 12515   WWalks cwwlk 25405   WWalksN cwwlkn 25406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-i2m1 9607  ax-1ne0 9608  ax-rrecex 9611  ax-cnre 9612
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-nn 10610  df-n0 10870  df-wwlkn 25408
This theorem is referenced by:  wwlknimp  25415  wwlkn0  25417  wlklniswwlkn1  25427  wlklniswwlkn2  25428  wwlkiswwlkn  25430  vfwlkniswwlkn  25434  wwlknred  25451  wwlknext  25452  wwlkextproplem3  25471  clwwlkel  25521  clwwlkf  25522  wwlksubclwwlk  25532  rusgranumwlkl1  25675  rusgra0edg  25683
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