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Theorem numclwwlkovf2 25824
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 463 . . 3  |-  ( ( V USGrph  E  /\  X  e.  V )  ->  X  e.  V )
2 2nn0 10893 . . 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 25821 . . 3  |-  ( ( X  e.  V  /\  2  e.  NN0 )  -> 
( X F 2 )  =  { w  e.  ( C `  2
)  |  ( w `
 0 )  =  X } )
61, 2, 5sylancl 669 . 2  |-  ( ( V USGrph  E  /\  X  e.  V )  ->  ( X F 2 )  =  { w  e.  ( C `  2 )  |  ( w ` 
0 )  =  X } )
73numclwwlkfvc 25817 . . . 4  |-  ( 2  e.  NN0  ->  ( C `
 2 )  =  ( ( V ClWWalksN  E ) `
 2 ) )
82, 7ax-mp 5 . . 3  |-  ( C `
 2 )  =  ( ( V ClWWalksN  E ) `
 2 )
9 rabeq 3040 . . 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 13 . 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 25515 . . . . . . 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 999 . . . . . . 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 265 . . . . . 6  |-  ( V USGrph  E  ->  ( w  e.  ( ( V ClWWalksN  E ) `
 2 )  <->  ( w  e. Word  V  /\  ( (
# `  w )  =  2  /\  {
( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) ) ) )
1413adantr 467 . . . . 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 712 . . . 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 655 . . . . 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 988 . . . . . 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 701 . . . . 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 256 . . . 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 265 . . 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 3036 . 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 2491 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 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   {crab 2743   {cpr 3972   class class class wbr 4405    |-> cmpt 4464   ran crn 4838   ` cfv 5585  (class class class)co 6295    |-> cmpt2 6297   0cc0 9544   1c1 9545   2c2 10666   NN0cn0 10876   #chash 12522  Word cword 12663   USGrph cusg 25069   ClWWalksN cclwwlkn 25489
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-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
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-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  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-int 4238  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-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12523  df-word 12671  df-lsw 12672  df-usgra 25072  df-clwwlk 25491  df-clwwlkn 25492
This theorem is referenced by:  numclwwlkovf2num  25825
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