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Theorem wwlkn 25422
Description: The set of walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Assertion
Ref Expression
wwlkn  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( ( V WWalksN  E
) `  N )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) } )
Distinct variable groups:    w, E    w, V    w, N
Allowed substitution hints:    X( w)    Y( w)

Proof of Theorem wwlkn
Dummy variables  e  n  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1011 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  ->  N  e.  NN0 )
2 ovex 6323 . . . 4  |-  ( V WWalks  E )  e.  _V
3 rabexg 4556 . . . 4  |-  ( ( V WWalks  E )  e. 
_V  ->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) }  e.  _V )
42, 3mp1i 13 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  ->  { w  e.  ( V WWalks  E )  |  (
# `  w )  =  ( N  + 
1 ) }  e.  _V )
5 oveq1 6302 . . . . . 6  |-  ( n  =  N  ->  (
n  +  1 )  =  ( N  + 
1 ) )
65eqeq2d 2463 . . . . 5  |-  ( n  =  N  ->  (
( # `  w )  =  ( n  + 
1 )  <->  ( # `  w
)  =  ( N  +  1 ) ) )
76rabbidv 3038 . . . 4  |-  ( n  =  N  ->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( n  +  1 ) }  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) } )
8 eqid 2453 . . . 4  |-  ( n  e.  NN0  |->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( n  +  1 ) } )  =  ( n  e.  NN0  |->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( n  +  1 ) } )
97, 8fvmptg 5951 . . 3  |-  ( ( N  e.  NN0  /\  { w  e.  ( V WWalks  E )  |  (
# `  w )  =  ( N  + 
1 ) }  e.  _V )  ->  ( ( n  e.  NN0  |->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( n  +  1 ) } ) `  N )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) } )
101, 4, 9syl2anc 667 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( n  +  1 ) } ) `  N )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) } )
11 df-wwlkn 25420 . . . . . . 7  |- WWalksN  =  ( v  e.  _V , 
e  e.  _V  |->  ( n  e.  NN0  |->  { w  e.  ( v WWalks  e )  |  ( # `  w
)  =  ( n  +  1 ) } ) )
1211a1i 11 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  -> WWalksN 
=  ( v  e. 
_V ,  e  e. 
_V  |->  ( n  e. 
NN0  |->  { w  e.  ( v WWalks  e )  |  ( # `  w
)  =  ( n  +  1 ) } ) ) )
13 oveq12 6304 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v WWalks  e )  =  ( V WWalks  E
) )
14 rabeq 3040 . . . . . . . . 9  |-  ( ( v WWalks  e )  =  ( V WWalks  E )  ->  { w  e.  ( v WWalks  e )  |  ( # `  w
)  =  ( n  +  1 ) }  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( n  +  1 ) } )
1513, 14syl 17 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  { w  e.  ( v WWalks  e )  |  ( # `  w
)  =  ( n  +  1 ) }  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( n  +  1 ) } )
1615mpteq2dv 4493 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( n  e.  NN0  |->  { w  e.  (
v WWalks  e )  |  (
# `  w )  =  ( n  + 
1 ) } )  =  ( n  e. 
NN0  |->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( n  +  1 ) } ) )
1716adantl 468 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
( n  e.  NN0  |->  { w  e.  (
v WWalks  e )  |  (
# `  w )  =  ( n  + 
1 ) } )  =  ( n  e. 
NN0  |->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( n  +  1 ) } ) )
18 elex 3056 . . . . . . 7  |-  ( V  e.  X  ->  V  e.  _V )
1918adantr 467 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  V  e.  _V )
20 elex 3056 . . . . . . 7  |-  ( E  e.  Y  ->  E  e.  _V )
2120adantl 468 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  E  e.  _V )
22 nn0ex 10882 . . . . . . . 8  |-  NN0  e.  _V
2322mptex 6141 . . . . . . 7  |-  ( n  e.  NN0  |->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( n  +  1 ) } )  e.  _V
2423a1i 11 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( n  e.  NN0  |->  { w  e.  ( V WWalks  E )  |  (
# `  w )  =  ( n  + 
1 ) } )  e.  _V )
2512, 17, 19, 21, 24ovmpt2d 6429 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V WWalksN  E )  =  ( n  e. 
NN0  |->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( n  +  1 ) } ) )
2625fveq1d 5872 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V WWalksN  E
) `  N )  =  ( ( n  e.  NN0  |->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( n  +  1 ) } ) `  N ) )
2726eqeq1d 2455 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( ( V WWalksN  E ) `  N
)  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) }  <-> 
( ( n  e. 
NN0  |->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( n  +  1 ) } ) `  N )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) } ) )
28273adant3 1029 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( ( ( V WWalksN  E ) `  N
)  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) }  <-> 
( ( n  e. 
NN0  |->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( n  +  1 ) } ) `  N )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) } ) )
2910, 28mpbird 236 1  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  -> 
( ( 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 986    = wceq 1446    e. wcel 1889   {crab 2743   _Vcvv 3047    |-> cmpt 4464   ` cfv 5585  (class class class)co 6295    |-> cmpt2 6297   1c1 9545    + caddc 9547   NN0cn0 10876   #chash 12522   WWalks cwwlk 25417   WWalksN cwwlkn 25418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-i2m1 9612  ax-1ne0 9613  ax-rrecex 9616  ax-cnre 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-nn 10617  df-n0 10877  df-wwlkn 25420
This theorem is referenced by:  iswwlkn  25424  wwlkn0s  25445  wwlknfi  25478
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