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Theorem numclwwlkovfel2 30825
Description: Properties of an element of the value of operation  F. (Contributed by Alexander van der Vekens, 20-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
numclwwlkovfel2  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( X F N )  <->  ( ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) { ( A `
 i ) ,  ( A `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E )  /\  ( # `  A
)  =  N  /\  ( A `  0 )  =  X ) ) )
Distinct variable groups:    n, E    n, N    n, V    i, E    i, N    i, V    w, i    w, C    w, N    C, n, v, w   
v, N    n, X, v, w    v, V    A, i, w
Allowed substitution hints:    A( v, n)    C( i)    E( w, v)    F( w, v, i, n)    V( w)    X( i)

Proof of Theorem numclwwlkovfel2
StepHypRef Expression
1 pm3.22 449 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  V )  ->  ( X  e.  V  /\  N  e.  NN0 ) )
213adant1 1006 . . . 4  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( X  e.  V  /\  N  e.  NN0 ) )
3 numclwwlk.c . . . . 5  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
4 numclwwlk.f . . . . 5  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
53, 4numclwwlkovf 30823 . . . 4  |-  ( ( X  e.  V  /\  N  e.  NN0 )  -> 
( X F N )  =  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X } )
62, 5syl 16 . . 3  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( X F N )  =  { w  e.  ( C `  N )  |  ( w ` 
0 )  =  X } )
76eleq2d 2524 . 2  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( X F N )  <->  A  e.  { w  e.  ( C `
 N )  |  ( w `  0
)  =  X }
) )
83numclwwlkfvc 30819 . . . . . . 7  |-  ( N  e.  NN0  ->  ( C `
 N )  =  ( ( V ClWWalksN  E ) `
 N ) )
983ad2ant2 1010 . . . . . 6  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( C `  N )  =  ( ( V ClWWalksN  E ) `  N
) )
109eleq2d 2524 . . . . 5  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( C `  N )  <->  A  e.  ( ( V ClWWalksN  E ) `
 N ) ) )
11 usgrav 23423 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
1211anim1i 568 . . . . . . . 8  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
13 df-3an 967 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
1412, 13sylibr 212 . . . . . . 7  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  ( V  e.  _V  /\  E  e.  _V  /\  N  e. 
NN0 ) )
15143adant3 1008 . . . . . 6  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( V  e.  _V  /\  E  e.  _V  /\  N  e. 
NN0 ) )
16 isclwwlkn 30581 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  ( A  e.  ( ( V ClWWalksN  E ) `  N
)  <->  ( A  e.  ( V ClWWalks  E )  /\  ( # `  A
)  =  N ) ) )
1715, 16syl 16 . . . . 5  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( ( V ClWWalksN  E ) `  N
)  <->  ( A  e.  ( V ClWWalks  E )  /\  ( # `  A
)  =  N ) ) )
18 isclwwlk 30580 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( A  e.  ( V ClWWalks  E )  <->  ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  A )  -  1 ) ) { ( A `  i ) ,  ( A `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E ) ) )
1911, 18syl 16 . . . . . . 7  |-  ( V USGrph  E  ->  ( A  e.  ( V ClWWalks  E )  <->  ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) { ( A `
 i ) ,  ( A `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E ) ) )
20193ad2ant1 1009 . . . . . 6  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( V ClWWalks  E )  <->  ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  A )  -  1 ) ) { ( A `  i ) ,  ( A `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E ) ) )
2120anbi1d 704 . . . . 5  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  (
( A  e.  ( V ClWWalks  E )  /\  ( # `
 A )  =  N )  <->  ( ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) { ( A `
 i ) ,  ( A `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E )  /\  ( # `  A
)  =  N ) ) )
2210, 17, 213bitrd 279 . . . 4  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( C `  N )  <->  ( ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) { ( A `
 i ) ,  ( A `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E )  /\  ( # `  A
)  =  N ) ) )
2322anbi1d 704 . . 3  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  (
( A  e.  ( C `  N )  /\  ( A ` 
0 )  =  X )  <->  ( ( ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) { ( A `
 i ) ,  ( A `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E )  /\  ( # `  A
)  =  N )  /\  ( A ` 
0 )  =  X ) ) )
24 fveq1 5799 . . . . 5  |-  ( w  =  A  ->  (
w `  0 )  =  ( A ` 
0 ) )
2524eqeq1d 2456 . . . 4  |-  ( w  =  A  ->  (
( w `  0
)  =  X  <->  ( A `  0 )  =  X ) )
2625elrab 3224 . . 3  |-  ( A  e.  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X }  <->  ( A  e.  ( C `  N
)  /\  ( A `  0 )  =  X ) )
27 df-3an 967 . . 3  |-  ( ( ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A )  -  1 ) ) { ( A `  i ) ,  ( A `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A ) ,  ( A `  0 ) }  e.  ran  E
)  /\  ( # `  A
)  =  N  /\  ( A `  0 )  =  X )  <->  ( (
( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A )  -  1 ) ) { ( A `  i ) ,  ( A `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A ) ,  ( A `  0 ) }  e.  ran  E
)  /\  ( # `  A
)  =  N )  /\  ( A ` 
0 )  =  X ) )
2823, 26, 273bitr4g 288 . 2  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X }  <->  ( ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) { ( A `
 i ) ,  ( A `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E )  /\  ( # `  A
)  =  N  /\  ( A `  0 )  =  X ) ) )
297, 28bitrd 253 1  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( X F N )  <->  ( ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) { ( A `
 i ) ,  ( A `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E )  /\  ( # `  A
)  =  N  /\  ( A `  0 )  =  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   {crab 2803   _Vcvv 3078   {cpr 3988   class class class wbr 4401    |-> cmpt 4459   ran crn 4950   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   0cc0 9394   1c1 9395    + caddc 9397    - cmin 9707   NN0cn0 10691  ..^cfzo 11666   #chash 12221  Word cword 12340   lastS clsw 12341   USGrph cusg 23417   ClWWalks cclwwlk 30562   ClWWalksN cclwwlkn 30563
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-word 12348  df-usgra 23419  df-clwwlk 30565  df-clwwlkn 30566
This theorem is referenced by:  numclwwlkovf2ex  30828  numclwlk1lem2foa  30833  numclwlk1lem2fo  30837
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