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Theorem rusgranumwlklem3 25169
Description: Lemma 3 for rusgranumwlk 25175. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
Hypotheses
Ref Expression
rusgranumwlk.w  |-  W  =  ( n  e.  NN0  |->  { c  e.  ( V Walks  E )  |  ( # `  ( 1st `  c ) )  =  n } )
rusgranumwlk.l  |-  L  =  ( v  e.  V ,  n  e.  NN0  |->  ( # `  { w  e.  ( W `  n
)  |  ( ( 2nd `  w ) `
 0 )  =  v } ) )
Assertion
Ref Expression
rusgranumwlklem3  |-  ( ( P  e.  V  /\  N  e.  NN0 )  -> 
( P L N )  =  ( # `  { w  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  w ) )  =  N  /\  (
( 2nd `  w
) `  0 )  =  P ) } ) )
Distinct variable groups:    E, c, n    N, c, n    V, c, n    v, N, w    P, n, v, w    v, V    n, W, v, w   
w, V, c
Allowed substitution hints:    P( c)    E( w, v)    L( w, v, n, c)    W( c)

Proof of Theorem rusgranumwlklem3
StepHypRef Expression
1 rusgranumwlk.w . . 3  |-  W  =  ( n  e.  NN0  |->  { c  e.  ( V Walks  E )  |  ( # `  ( 1st `  c ) )  =  n } )
2 rusgranumwlk.l . . 3  |-  L  =  ( v  e.  V ,  n  e.  NN0  |->  ( # `  { w  e.  ( W `  n
)  |  ( ( 2nd `  w ) `
 0 )  =  v } ) )
31, 2rusgranumwlklem2 25168 . 2  |-  ( ( P  e.  V  /\  N  e.  NN0 )  -> 
( P L N )  =  ( # `  { w  e.  ( W `  N )  |  ( ( 2nd `  w ) `  0
)  =  P }
) )
4 eqeq2 2472 . . . . . . . . . . 11  |-  ( n  =  N  ->  (
( # `  ( 1st `  c ) )  =  n  <->  ( # `  ( 1st `  c ) )  =  N ) )
54rabbidv 3101 . . . . . . . . . 10  |-  ( n  =  N  ->  { c  e.  ( V Walks  E
)  |  ( # `  ( 1st `  c
) )  =  n }  =  { c  e.  ( V Walks  E
)  |  ( # `  ( 1st `  c
) )  =  N } )
6 ovex 6324 . . . . . . . . . . 11  |-  ( V Walks 
E )  e.  _V
76rabex 4607 . . . . . . . . . 10  |-  { c  e.  ( V Walks  E
)  |  ( # `  ( 1st `  c
) )  =  N }  e.  _V
85, 1, 7fvmpt 5956 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( W `
 N )  =  { c  e.  ( V Walks  E )  |  ( # `  ( 1st `  c ) )  =  N } )
98eleq2d 2527 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( w  e.  ( W `  N )  <->  w  e.  { c  e.  ( V Walks 
E )  |  (
# `  ( 1st `  c ) )  =  N } ) )
10 fveq2 5872 . . . . . . . . . . 11  |-  ( c  =  w  ->  ( 1st `  c )  =  ( 1st `  w
) )
1110fveq2d 5876 . . . . . . . . . 10  |-  ( c  =  w  ->  ( # `
 ( 1st `  c
) )  =  (
# `  ( 1st `  w ) ) )
1211eqeq1d 2459 . . . . . . . . 9  |-  ( c  =  w  ->  (
( # `  ( 1st `  c ) )  =  N  <->  ( # `  ( 1st `  w ) )  =  N ) )
1312elrab 3257 . . . . . . . 8  |-  ( w  e.  { c  e.  ( V Walks  E )  |  ( # `  ( 1st `  c ) )  =  N }  <->  ( w  e.  ( V Walks  E )  /\  ( # `  ( 1st `  w ) )  =  N ) )
149, 13syl6bb 261 . . . . . . 7  |-  ( N  e.  NN0  ->  ( w  e.  ( W `  N )  <->  ( w  e.  ( V Walks  E )  /\  ( # `  ( 1st `  w ) )  =  N ) ) )
1514adantl 466 . . . . . 6  |-  ( ( P  e.  V  /\  N  e.  NN0 )  -> 
( w  e.  ( W `  N )  <-> 
( w  e.  ( V Walks  E )  /\  ( # `  ( 1st `  w ) )  =  N ) ) )
1615anbi1d 704 . . . . 5  |-  ( ( P  e.  V  /\  N  e.  NN0 )  -> 
( ( w  e.  ( W `  N
)  /\  ( ( 2nd `  w ) ` 
0 )  =  P )  <->  ( ( w  e.  ( V Walks  E
)  /\  ( # `  ( 1st `  w ) )  =  N )  /\  ( ( 2nd `  w
) `  0 )  =  P ) ) )
17 anass 649 . . . . 5  |-  ( ( ( w  e.  ( V Walks  E )  /\  ( # `  ( 1st `  w ) )  =  N )  /\  (
( 2nd `  w
) `  0 )  =  P )  <->  ( w  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  w
) )  =  N  /\  ( ( 2nd `  w ) `  0
)  =  P ) ) )
1816, 17syl6bb 261 . . . 4  |-  ( ( P  e.  V  /\  N  e.  NN0 )  -> 
( ( w  e.  ( W `  N
)  /\  ( ( 2nd `  w ) ` 
0 )  =  P )  <->  ( w  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  w
) )  =  N  /\  ( ( 2nd `  w ) `  0
)  =  P ) ) ) )
1918rabbidva2 3099 . . 3  |-  ( ( P  e.  V  /\  N  e.  NN0 )  ->  { w  e.  ( W `  N )  |  ( ( 2nd `  w ) `  0
)  =  P }  =  { w  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  w ) )  =  N  /\  (
( 2nd `  w
) `  0 )  =  P ) } )
2019fveq2d 5876 . 2  |-  ( ( P  e.  V  /\  N  e.  NN0 )  -> 
( # `  { w  e.  ( W `  N
)  |  ( ( 2nd `  w ) `
 0 )  =  P } )  =  ( # `  {
w  e.  ( V Walks 
E )  |  ( ( # `  ( 1st `  w ) )  =  N  /\  (
( 2nd `  w
) `  0 )  =  P ) } ) )
213, 20eqtrd 2498 1  |-  ( ( P  e.  V  /\  N  e.  NN0 )  -> 
( P L N )  =  ( # `  { w  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  w ) )  =  N  /\  (
( 2nd `  w
) `  0 )  =  P ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   {crab 2811    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798   0cc0 9509   NN0cn0 10816   #chash 12408   Walks cwalk 24716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301
This theorem is referenced by:  rusgranumwlklem4  25170  rusgranumwlkg  25176
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