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Theorem rusgranumwlklem4 30710
Description: Lemma 4 for rusgranumwlk 30715. (Contributed by Alexander van der Vekens, 24-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
rusgranumwlklem4  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  ( P L N )  =  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( 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    v, E, w
Allowed substitution hints:    P( c)    L( w, v, n, c)    W( c)

Proof of Theorem rusgranumwlklem4
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 rusgranumwlk.w . . . 4  |-  W  =  ( n  e.  NN0  |->  { c  e.  ( V Walks  E )  |  ( # `  ( 1st `  c ) )  =  n } )
2 rusgranumwlk.l . . . 4  |-  L  =  ( v  e.  V ,  n  e.  NN0  |->  ( # `  { w  e.  ( W `  n
)  |  ( ( 2nd `  w ) `
 0 )  =  v } ) )
31, 2rusgranumwlklem3 30709 . . 3  |-  ( ( P  e.  V  /\  N  e.  NN0 )  -> 
( P L N )  =  ( # `  { w  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  w ) )  =  N  /\  (
( 2nd `  w
) `  0 )  =  P ) } ) )
433adant1 1006 . 2  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  ( P L N )  =  ( # `  {
w  e.  ( V Walks 
E )  |  ( ( # `  ( 1st `  w ) )  =  N  /\  (
( 2nd `  w
) `  0 )  =  P ) } ) )
5 fveq2 5791 . . . . . . . 8  |-  ( w  =  v  ->  ( 1st `  w )  =  ( 1st `  v
) )
65fveq2d 5795 . . . . . . 7  |-  ( w  =  v  ->  ( # `
 ( 1st `  w
) )  =  (
# `  ( 1st `  v ) ) )
76eqeq1d 2453 . . . . . 6  |-  ( w  =  v  ->  (
( # `  ( 1st `  w ) )  =  N  <->  ( # `  ( 1st `  v ) )  =  N ) )
8 fveq2 5791 . . . . . . . 8  |-  ( w  =  v  ->  ( 2nd `  w )  =  ( 2nd `  v
) )
98fveq1d 5793 . . . . . . 7  |-  ( w  =  v  ->  (
( 2nd `  w
) `  0 )  =  ( ( 2nd `  v ) `  0
) )
109eqeq1d 2453 . . . . . 6  |-  ( w  =  v  ->  (
( ( 2nd `  w
) `  0 )  =  P  <->  ( ( 2nd `  v ) `  0
)  =  P ) )
117, 10anbi12d 710 . . . . 5  |-  ( w  =  v  ->  (
( ( # `  ( 1st `  w ) )  =  N  /\  (
( 2nd `  w
) `  0 )  =  P )  <->  ( ( # `
 ( 1st `  v
) )  =  N  /\  ( ( 2nd `  v ) `  0
)  =  P ) ) )
1211cbvrabv 3069 . . . 4  |-  { w  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  w
) )  =  N  /\  ( ( 2nd `  w ) `  0
)  =  P ) }  =  { v  e.  ( V Walks  E
)  |  ( (
# `  ( 1st `  v ) )  =  N  /\  ( ( 2nd `  v ) `
 0 )  =  P ) }
1312a1i 11 . . 3  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  { w  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  w
) )  =  N  /\  ( ( 2nd `  w ) `  0
)  =  P ) }  =  { v  e.  ( V Walks  E
)  |  ( (
# `  ( 1st `  v ) )  =  N  /\  ( ( 2nd `  v ) `
 0 )  =  P ) } )
1413fveq2d 5795 . 2  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  ( # `
 { w  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  w
) )  =  N  /\  ( ( 2nd `  w ) `  0
)  =  P ) } )  =  (
# `  { v  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  v
) )  =  N  /\  ( ( 2nd `  v ) `  0
)  =  P ) } ) )
15 fvex 5801 . . . . . 6  |-  ( ( V WWalksN  E ) `  N
)  e.  _V
1615rabex 4543 . . . . 5  |-  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  e.  _V
1716a1i 11 . . . 4  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  e.  _V )
18 ovex 6217 . . . . 5  |-  ( V Walks 
E )  e.  _V
1918rabex 4543 . . . 4  |-  { v  e.  ( V Walks  E
)  |  ( (
# `  ( 1st `  v ) )  =  N  /\  ( ( 2nd `  v ) `
 0 )  =  P ) }  e.  _V
2017, 19jctil 537 . . 3  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  ( { v  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  v ) )  =  N  /\  (
( 2nd `  v
) `  0 )  =  P ) }  e.  _V  /\  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  e.  _V )
)
21 wlkiswwlkbij2 30493 . . 3  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  E. f 
f : { v  e.  ( V Walks  E
)  |  ( (
# `  ( 1st `  v ) )  =  N  /\  ( ( 2nd `  v ) `
 0 )  =  P ) } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P } )
22 hasheqf1oi 12225 . . 3  |-  ( ( { v  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  v ) )  =  N  /\  (
( 2nd `  v
) `  0 )  =  P ) }  e.  _V  /\  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  e.  _V )  ->  ( E. f  f : { v  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  v
) )  =  N  /\  ( ( 2nd `  v ) `  0
)  =  P ) } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P }  ->  (
# `  { v  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  v
) )  =  N  /\  ( ( 2nd `  v ) `  0
)  =  P ) } )  =  (
# `  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P } ) ) )
2320, 21, 22sylc 60 . 2  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  ( # `
 { v  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  v
) )  =  N  /\  ( ( 2nd `  v ) `  0
)  =  P ) } )  =  (
# `  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P } ) )
244, 14, 233eqtrd 2496 1  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  ( P L N )  =  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   {crab 2799   _Vcvv 3070   class class class wbr 4392    |-> cmpt 4450   -1-1-onto->wf1o 5517   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   1stc1st 6677   2ndc2nd 6678   0cc0 9385   NN0cn0 10682   #chash 12206   USGrph cusg 23401   Walks cwalk 23542   WWalksN cwwlkn 30452
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-card 8212  df-cda 8440  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-fzo 11652  df-hash 12207  df-word 12333  df-usgra 23403  df-wlk 23552  df-wwlk 30453  df-wwlkn 30454
This theorem is referenced by:  rusgranumwlkb0  30711  rusgranumwlkb1  30712  rusgra0edg  30713  rusgranumwlks  30714
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