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Theorem numclwwlkovf2 25302
Description: Value of operation  F for argument 2. (Contributed by Alexander van der Vekens, 19-Sep-2018.)
Hypotheses
Ref Expression
numclwwlk.c  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
numclwwlk.f  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
Assertion
Ref Expression
numclwwlkovf2  |-  ( ( V USGrph  E  /\  X  e.  V )  ->  ( X F 2 )  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E  /\  ( w ` 
0 )  =  X ) } )
Distinct variable groups:    n, E    n, V    w, C, n, v    n, X, v, w    v, V    w, E    w, V
Allowed substitution hints:    E( v)    F( w, v, n)

Proof of Theorem numclwwlkovf2
StepHypRef Expression
1 simpr 461 . . 3  |-  ( ( V USGrph  E  /\  X  e.  V )  ->  X  e.  V )
2 2nn0 10833 . . 3  |-  2  e.  NN0
3 numclwwlk.c . . . 4  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
4 numclwwlk.f . . . 4  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
53, 4numclwwlkovf 25299 . . 3  |-  ( ( X  e.  V  /\  2  e.  NN0 )  -> 
( X F 2 )  =  { w  e.  ( C `  2
)  |  ( w `
 0 )  =  X } )
61, 2, 5sylancl 662 . 2  |-  ( ( V USGrph  E  /\  X  e.  V )  ->  ( X F 2 )  =  { w  e.  ( C `  2 )  |  ( w ` 
0 )  =  X } )
73numclwwlkfvc 25295 . . . 4  |-  ( 2  e.  NN0  ->  ( C `
 2 )  =  ( ( V ClWWalksN  E ) `
 2 ) )
82, 7ax-mp 5 . . 3  |-  ( C `
 2 )  =  ( ( V ClWWalksN  E ) `
 2 )
9 rabeq 3103 . . 3  |-  ( ( C `  2 )  =  ( ( V ClWWalksN  E ) `  2
)  ->  { w  e.  ( C `  2
)  |  ( w `
 0 )  =  X }  =  {
w  e.  ( ( V ClWWalksN  E ) `  2
)  |  ( w `
 0 )  =  X } )
108, 9mp1i 12 . 2  |-  ( ( V USGrph  E  /\  X  e.  V )  ->  { w  e.  ( C `  2
)  |  ( w `
 0 )  =  X }  =  {
w  e.  ( ( V ClWWalksN  E ) `  2
)  |  ( w `
 0 )  =  X } )
11 clwwlkn2 24993 . . . . . . 7  |-  ( V USGrph  E  ->  ( w  e.  ( ( V ClWWalksN  E ) `
 2 )  <->  ( ( # `
 w )  =  2  /\  w  e. Word  V  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E ) ) )
12 3anan12 986 . . . . . . 7  |-  ( ( ( # `  w
)  =  2  /\  w  e. Word  V  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E )  <-> 
( w  e. Word  V  /\  ( ( # `  w
)  =  2  /\ 
{ ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E
) ) )
1311, 12syl6bb 261 . . . . . 6  |-  ( V USGrph  E  ->  ( w  e.  ( ( V ClWWalksN  E ) `
 2 )  <->  ( w  e. Word  V  /\  ( (
# `  w )  =  2  /\  {
( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) ) ) )
1413adantr 465 . . . . 5  |-  ( ( V USGrph  E  /\  X  e.  V )  ->  (
w  e.  ( ( V ClWWalksN  E ) `  2
)  <->  ( w  e. Word  V  /\  ( ( # `  w )  =  2  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E ) ) ) )
1514anbi1d 704 . . . 4  |-  ( ( V USGrph  E  /\  X  e.  V )  ->  (
( w  e.  ( ( V ClWWalksN  E ) `  2 )  /\  ( w `  0
)  =  X )  <-> 
( ( w  e. Word  V  /\  ( ( # `  w )  =  2  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E ) )  /\  (
w `  0 )  =  X ) ) )
16 anass 649 . . . . 5  |-  ( ( ( w  e. Word  V  /\  ( ( # `  w
)  =  2  /\ 
{ ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E
) )  /\  (
w `  0 )  =  X )  <->  ( w  e. Word  V  /\  ( ( ( # `  w
)  =  2  /\ 
{ ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E
)  /\  ( w `  0 )  =  X ) ) )
17 df-3an 975 . . . . . 6  |-  ( ( ( # `  w
)  =  2  /\ 
{ ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E  /\  ( w `  0
)  =  X )  <-> 
( ( ( # `  w )  =  2  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E )  /\  ( w `
 0 )  =  X ) )
1817anbi2i 694 . . . . 5  |-  ( ( w  e. Word  V  /\  ( ( # `  w
)  =  2  /\ 
{ ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E  /\  ( w `  0
)  =  X ) )  <->  ( w  e. Word  V  /\  ( ( (
# `  w )  =  2  /\  {
( w `  0
) ,  ( w `
 1 ) }  e.  ran  E )  /\  ( w ` 
0 )  =  X ) ) )
1916, 18bitr4i 252 . . . 4  |-  ( ( ( w  e. Word  V  /\  ( ( # `  w
)  =  2  /\ 
{ ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E
) )  /\  (
w `  0 )  =  X )  <->  ( w  e. Word  V  /\  ( (
# `  w )  =  2  /\  {
( w `  0
) ,  ( w `
 1 ) }  e.  ran  E  /\  ( w `  0
)  =  X ) ) )
2015, 19syl6bb 261 . . 3  |-  ( ( V USGrph  E  /\  X  e.  V )  ->  (
( w  e.  ( ( V ClWWalksN  E ) `  2 )  /\  ( w `  0
)  =  X )  <-> 
( w  e. Word  V  /\  ( ( # `  w
)  =  2  /\ 
{ ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E  /\  ( w `  0
)  =  X ) ) ) )
2120rabbidva2 3099 . 2  |-  ( ( V USGrph  E  /\  X  e.  V )  ->  { w  e.  ( ( V ClWWalksN  E ) `
 2 )  |  ( w `  0
)  =  X }  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E  /\  ( w ` 
0 )  =  X ) } )
226, 10, 213eqtrd 2502 1  |-  ( ( V USGrph  E  /\  X  e.  V )  ->  ( X F 2 )  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E  /\  ( w ` 
0 )  =  X ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {crab 2811   {cpr 4034   class class class wbr 4456    |-> cmpt 4515   ran crn 5009   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   0cc0 9509   1c1 9510   2c2 10606   NN0cn0 10816   #chash 12408  Word cword 12538   USGrph cusg 24548   ClWWalksN cclwwlkn 24967
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-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-lsw 12547  df-usgra 24551  df-clwwlk 24969  df-clwwlkn 24970
This theorem is referenced by:  numclwwlkovf2num  25303
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