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Theorem rusgranumwlklem4 25154
Description: Lemma 4 for rusgranumwlk 25159. (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 25153 . . 3  |-  ( ( P  e.  V  /\  N  e.  NN0 )  -> 
( P L N )  =  ( # `  { w  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  w ) )  =  N  /\  (
( 2nd `  w
) `  0 )  =  P ) } ) )
433adant1 1012 . 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 5848 . . . . . . . 8  |-  ( w  =  v  ->  ( 1st `  w )  =  ( 1st `  v
) )
65fveq2d 5852 . . . . . . 7  |-  ( w  =  v  ->  ( # `
 ( 1st `  w
) )  =  (
# `  ( 1st `  v ) ) )
76eqeq1d 2456 . . . . . 6  |-  ( w  =  v  ->  (
( # `  ( 1st `  w ) )  =  N  <->  ( # `  ( 1st `  v ) )  =  N ) )
8 fveq2 5848 . . . . . . . 8  |-  ( w  =  v  ->  ( 2nd `  w )  =  ( 2nd `  v
) )
98fveq1d 5850 . . . . . . 7  |-  ( w  =  v  ->  (
( 2nd `  w
) `  0 )  =  ( ( 2nd `  v ) `  0
) )
109eqeq1d 2456 . . . . . 6  |-  ( w  =  v  ->  (
( ( 2nd `  w
) `  0 )  =  P  <->  ( ( 2nd `  v ) `  0
)  =  P ) )
117, 10anbi12d 708 . . . . 5  |-  ( w  =  v  ->  (
( ( # `  ( 1st `  w ) )  =  N  /\  (
( 2nd `  w
) `  0 )  =  P )  <->  ( ( # `
 ( 1st `  v
) )  =  N  /\  ( ( 2nd `  v ) `  0
)  =  P ) ) )
1211cbvrabv 3105 . . . 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 5852 . 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 5858 . . . . . 6  |-  ( ( V WWalksN  E ) `  N
)  e.  _V
1615rabex 4588 . . . . 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 6298 . . . . 5  |-  ( V Walks 
E )  e.  _V
1918rabex 4588 . . . 4  |-  { v  e.  ( V Walks  E
)  |  ( (
# `  ( 1st `  v ) )  =  N  /\  ( ( 2nd `  v ) `
 0 )  =  P ) }  e.  _V
2017, 19jctil 535 . . 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 24923 . . 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 12406 . . 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 2499 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 367    /\ w3a 971    = wceq 1398   E.wex 1617    e. wcel 1823   {crab 2808   _Vcvv 3106   class class class wbr 4439    |-> cmpt 4497   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   1stc1st 6771   2ndc2nd 6772   0cc0 9481   NN0cn0 10791   #chash 12387   USGrph cusg 24532   Walks cwalk 24700   WWalksN cwwlkn 24880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-usgra 24535  df-wlk 24710  df-wwlk 24881  df-wwlkn 24882
This theorem is referenced by:  rusgranumwlkb0  25155  rusgranumwlkb1  25156  rusgra0edg  25157  rusgranumwlks  25158
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