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Theorem wwlkextprop 24567
Description: Adding additional properties to the set of walks (as words) of a fixed length starting at a fixed vertex. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
Hypotheses
Ref Expression
wwlkextprop.x  |-  X  =  ( ( V WWalksN  E
) `  ( N  +  1 ) )
wwlkextprop.y  |-  Y  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P }
Assertion
Ref Expression
wwlkextprop  |-  ( N  e.  NN0  ->  { x  e.  X  |  (
x `  0 )  =  P }  =  {
x  e.  X  |  E. y  e.  Y  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E ) } )
Distinct variable groups:    w, E    w, N    w, P    w, V    y, E    x, N, y, w    y, P    y, X    y, Y
Allowed substitution hints:    P( x)    E( x)    V( x, y)    X( x, w)    Y( x, w)

Proof of Theorem wwlkextprop
StepHypRef Expression
1 eqidd 2468 . . . . 5  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  ( x substr  <.
0 ,  ( N  +  1 ) >.
)  =  ( x substr  <. 0 ,  ( N  +  1 ) >.
) )
2 wwlkextprop.x . . . . . . . . 9  |-  X  =  ( ( V WWalksN  E
) `  ( N  +  1 ) )
32wwlkextproplem1 24564 . . . . . . . 8  |-  ( ( x  e.  X  /\  N  e.  NN0 )  -> 
( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  ( x `  0
) )
43ancoms 453 . . . . . . 7  |-  ( ( N  e.  NN0  /\  x  e.  X )  ->  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  ( x `  0
) )
54adantr 465 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  ( (
x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 )  =  ( x ` 
0 ) )
6 eqeq2 2482 . . . . . . 7  |-  ( ( x `  0 )  =  P  ->  (
( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  ( x `  0
)  <->  ( ( x substr  <. 0 ,  ( N  +  1 ) >.
) `  0 )  =  P ) )
76adantl 466 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  ( (
( x substr  <. 0 ,  ( N  +  1 ) >. ) `  0
)  =  ( x `
 0 )  <->  ( (
x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 )  =  P ) )
85, 7mpbid 210 . . . . 5  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  ( (
x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 )  =  P )
92wwlkextproplem2 24565 . . . . . . 7  |-  ( ( x  e.  X  /\  N  e.  NN0 )  ->  { ( lastS  `  ( x substr  <. 0 ,  ( N  +  1 ) >.
) ) ,  ( lastS  `  x ) }  e.  ran  E )
109ancoms 453 . . . . . 6  |-  ( ( N  e.  NN0  /\  x  e.  X )  ->  { ( lastS  `  (
x substr  <. 0 ,  ( N  +  1 )
>. ) ) ,  ( lastS  `  x ) }  e.  ran  E )
1110adantr 465 . . . . 5  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  { ( lastS  `  ( x substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  x
) }  e.  ran  E )
12 simpr 461 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  x  e.  X )  ->  x  e.  X )
1312adantr 465 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  x  e.  X )
14 simpr 461 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  ( x `  0 )  =  P )
15 simpll 753 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  N  e.  NN0 )
16 wwlkextprop.y . . . . . . . 8  |-  Y  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P }
172, 16wwlkextproplem3 24566 . . . . . . 7  |-  ( ( x  e.  X  /\  ( x `  0
)  =  P  /\  N  e.  NN0 )  -> 
( x substr  <. 0 ,  ( N  +  1 ) >. )  e.  Y
)
1813, 14, 15, 17syl3anc 1228 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  ( x substr  <.
0 ,  ( N  +  1 ) >.
)  e.  Y )
19 eqeq2 2482 . . . . . . . 8  |-  ( y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  <->  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )
) )
20 fveq1 5871 . . . . . . . . 9  |-  ( y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( y `  0
)  =  ( ( x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 ) )
2120eqeq1d 2469 . . . . . . . 8  |-  ( y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( ( y ` 
0 )  =  P  <-> 
( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  P ) )
22 fveq2 5872 . . . . . . . . . 10  |-  ( y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( lastS  `  y )  =  ( lastS  `  ( x substr  <. 0 ,  ( N  +  1 ) >.
) ) )
2322preq1d 4118 . . . . . . . . 9  |-  ( y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  ->  { ( lastS  `  y
) ,  ( lastS  `  x
) }  =  {
( lastS  `  ( x substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  x
) } )
2423eleq1d 2536 . . . . . . . 8  |-  ( y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( { ( lastS  `  y
) ,  ( lastS  `  x
) }  e.  ran  E  <->  { ( lastS  `  ( x substr  <. 0 ,  ( N  +  1 ) >.
) ) ,  ( lastS  `  x ) }  e.  ran  E ) )
2519, 21, 243anbi123d 1299 . . . . . . 7  |-  ( y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( ( ( x substr  <. 0 ,  ( N  +  1 ) >.
)  =  y  /\  ( y `  0
)  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E
)  <->  ( ( x substr  <. 0 ,  ( N  +  1 ) >.
)  =  ( x substr  <. 0 ,  ( N  +  1 ) >.
)  /\  ( (
x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 )  =  P  /\  {
( lastS  `  ( x substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )
2625adantl 466 . . . . . 6  |-  ( ( ( ( N  e. 
NN0  /\  x  e.  X )  /\  (
x `  0 )  =  P )  /\  y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )
)  ->  ( (
( x substr  <. 0 ,  ( N  +  1 ) >. )  =  y  /\  ( y ` 
0 )  =  P  /\  { ( lastS  `  y
) ,  ( lastS  `  x
) }  e.  ran  E )  <->  ( ( x substr  <. 0 ,  ( N  +  1 ) >.
)  =  ( x substr  <. 0 ,  ( N  +  1 ) >.
)  /\  ( (
x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 )  =  P  /\  {
( lastS  `  ( x substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )
2718, 26rspcedv 3223 . . . . 5  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  ( (
( x substr  <. 0 ,  ( N  +  1 ) >. )  =  ( x substr  <. 0 ,  ( N  +  1 )
>. )  /\  (
( x substr  <. 0 ,  ( N  +  1 ) >. ) `  0
)  =  P  /\  { ( lastS  `  ( x substr  <.
0 ,  ( N  +  1 ) >.
) ) ,  ( lastS  `  x ) }  e.  ran  E )  ->  E. y  e.  Y  ( (
x substr  <. 0 ,  ( N  +  1 )
>. )  =  y  /\  ( y `  0
)  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E
) ) )
281, 8, 11, 27mp3and 1327 . . . 4  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  E. y  e.  Y  ( (
x substr  <. 0 ,  ( N  +  1 )
>. )  =  y  /\  ( y `  0
)  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E
) )
2928ex 434 . . 3  |-  ( ( N  e.  NN0  /\  x  e.  X )  ->  ( ( x ` 
0 )  =  P  ->  E. y  e.  Y  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E ) ) )
3020eqcoms 2479 . . . . . . . . 9  |-  ( ( x substr  <. 0 ,  ( N  +  1 )
>. )  =  y  ->  ( y `  0
)  =  ( ( x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 ) )
3130eqeq1d 2469 . . . . . . . 8  |-  ( ( x substr  <. 0 ,  ( N  +  1 )
>. )  =  y  ->  ( ( y ` 
0 )  =  P  <-> 
( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  P ) )
323eqcomd 2475 . . . . . . . . . . 11  |-  ( ( x  e.  X  /\  N  e.  NN0 )  -> 
( x `  0
)  =  ( ( x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 ) )
3332ancoms 453 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  x  e.  X )  ->  ( x `  0
)  =  ( ( x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 ) )
3433adantr 465 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  y  e.  Y
)  ->  ( x `  0 )  =  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 ) )
35 eqeq2 2482 . . . . . . . . . 10  |-  ( P  =  ( ( x substr  <. 0 ,  ( N  +  1 ) >.
) `  0 )  ->  ( ( x ` 
0 )  =  P  <-> 
( x `  0
)  =  ( ( x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 ) ) )
3635eqcoms 2479 . . . . . . . . 9  |-  ( ( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  P  ->  ( (
x `  0 )  =  P  <->  ( x ` 
0 )  =  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 ) ) )
3734, 36syl5ibr 221 . . . . . . . 8  |-  ( ( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  P  ->  ( (
( N  e.  NN0  /\  x  e.  X )  /\  y  e.  Y
)  ->  ( x `  0 )  =  P ) )
3831, 37syl6bi 228 . . . . . . 7  |-  ( ( x substr  <. 0 ,  ( N  +  1 )
>. )  =  y  ->  ( ( y ` 
0 )  =  P  ->  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  y  e.  Y
)  ->  ( x `  0 )  =  P ) ) )
3938imp 429 . . . . . 6  |-  ( ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P )  ->  (
( ( N  e. 
NN0  /\  x  e.  X )  /\  y  e.  Y )  ->  (
x `  0 )  =  P ) )
40393adant3 1016 . . . . 5  |-  ( ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E )  ->  (
( ( N  e. 
NN0  /\  x  e.  X )  /\  y  e.  Y )  ->  (
x `  0 )  =  P ) )
4140com12 31 . . . 4  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  y  e.  Y
)  ->  ( (
( x substr  <. 0 ,  ( N  +  1 ) >. )  =  y  /\  ( y ` 
0 )  =  P  /\  { ( lastS  `  y
) ,  ( lastS  `  x
) }  e.  ran  E )  ->  ( x `  0 )  =  P ) )
4241rexlimdva 2959 . . 3  |-  ( ( N  e.  NN0  /\  x  e.  X )  ->  ( E. y  e.  Y  ( ( x substr  <. 0 ,  ( N  +  1 ) >.
)  =  y  /\  ( y `  0
)  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E
)  ->  ( x `  0 )  =  P ) )
4329, 42impbid 191 . 2  |-  ( ( N  e.  NN0  /\  x  e.  X )  ->  ( ( x ` 
0 )  =  P  <->  E. y  e.  Y  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E ) ) )
4443rabbidva 3109 1  |-  ( N  e.  NN0  ->  { x  e.  X  |  (
x `  0 )  =  P }  =  {
x  e.  X  |  E. y  e.  Y  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2818   {crab 2821   {cpr 4035   <.cop 4039   ran crn 5006   ` cfv 5594  (class class class)co 6295   0cc0 9504   1c1 9505    + caddc 9507   NN0cn0 10807   lastS clsw 12516   substr csubstr 12519   WWalksN cwwlkn 24501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-lsw 12524  df-substr 12527  df-wwlk 24502  df-wwlkn 24503
This theorem is referenced by: (None)
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