MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wwlkextbij Structured version   Unicode version

Theorem wwlkextbij 24938
Description: There is a bijection between the extensions of a walk (as word) by an edge and the set of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 21-Aug-2018.)
Assertion
Ref Expression
wwlkextbij  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  E. f 
f : { w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) } -1-1-onto-> { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E } )
Distinct variable groups:    f, E, w    f, N, w    f, V, w    f, W, w   
n, E    n, V    n, W, f
Allowed substitution hint:    N( n)

Proof of Theorem wwlkextbij
Dummy variables  t  x  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5858 . . . 4  |-  ( ( V WWalksN  E ) `  ( N  +  1 ) )  e.  _V
21a1i 11 . . 3  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V WWalksN  E ) `  ( N  +  1 ) )  e.  _V )
3 rabexg 4587 . . 3  |-  ( ( ( V WWalksN  E ) `  ( N  +  1 ) )  e.  _V  ->  { t  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( t substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E ) }  e.  _V )
4 mptexg 6117 . . 3  |-  ( { t  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( t substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) }  e.  _V  ->  ( x  e.  {
t  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( t substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) }  |->  ( lastS  `  x
) )  e.  _V )
52, 3, 43syl 20 . 2  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( x  e.  { t  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( t substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E ) }  |->  ( lastS  `  x ) )  e. 
_V )
6 eqid 2454 . . . 4  |-  { w  e. Word  V  |  ( (
# `  w )  =  ( N  + 
2 )  /\  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) }  =  {
w  e. Word  V  | 
( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) }
7 preq2 4096 . . . . . 6  |-  ( n  =  p  ->  { ( lastS  `  W ) ,  n }  =  { ( lastS  `  W ) ,  p } )
87eleq1d 2523 . . . . 5  |-  ( n  =  p  ->  ( { ( lastS  `  W ) ,  n }  e.  ran  E  <->  { ( lastS  `  W
) ,  p }  e.  ran  E ) )
98cbvrabv 3105 . . . 4  |-  { n  e.  V  |  {
( lastS  `  W ) ,  n }  e.  ran  E }  =  { p  e.  V  |  {
( lastS  `  W ) ,  p }  e.  ran  E }
10 fveq2 5848 . . . . . . . 8  |-  ( t  =  w  ->  ( # `
 t )  =  ( # `  w
) )
1110eqeq1d 2456 . . . . . . 7  |-  ( t  =  w  ->  (
( # `  t )  =  ( N  + 
2 )  <->  ( # `  w
)  =  ( N  +  2 ) ) )
12 oveq1 6277 . . . . . . . 8  |-  ( t  =  w  ->  (
t substr  <. 0 ,  ( N  +  1 )
>. )  =  (
w substr  <. 0 ,  ( N  +  1 )
>. ) )
1312eqeq1d 2456 . . . . . . 7  |-  ( t  =  w  ->  (
( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  <-> 
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
14 fveq2 5848 . . . . . . . . 9  |-  ( t  =  w  ->  ( lastS  `  t )  =  ( lastS  `  w ) )
1514preq2d 4102 . . . . . . . 8  |-  ( t  =  w  ->  { ( lastS  `  W ) ,  ( lastS  `  t ) }  =  { ( lastS  `  W ) ,  ( lastS  `  w
) } )
1615eleq1d 2523 . . . . . . 7  |-  ( t  =  w  ->  ( { ( lastS  `  W ) ,  ( lastS  `  t
) }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( lastS  `  w
) }  e.  ran  E ) )
1711, 13, 163anbi123d 1297 . . . . . 6  |-  ( t  =  w  ->  (
( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E )  <->  ( ( # `  w )  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) ) )
1817cbvrabv 3105 . . . . 5  |-  { t  e. Word  V  |  ( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) }  =  {
w  e. Word  V  | 
( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) }
19 mpteq1 4519 . . . . 5  |-  ( { t  e. Word  V  | 
( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) }  =  {
w  e. Word  V  | 
( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) }  ->  (
x  e.  { t  e. Word  V  |  ( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) }  |->  ( lastS  `  x
) )  =  ( x  e.  { w  e. Word  V  |  ( (
# `  w )  =  ( N  + 
2 )  /\  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) }  |->  ( lastS  `  x
) ) )
2018, 19ax-mp 5 . . . 4  |-  ( x  e.  { t  e. Word  V  |  ( ( # `
 t )  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E
) }  |->  ( lastS  `  x
) )  =  ( x  e.  { w  e. Word  V  |  ( (
# `  w )  =  ( N  + 
2 )  /\  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) }  |->  ( lastS  `  x
) )
216, 9, 20wwlkextbij0 24937 . . 3  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( x  e.  { t  e. Word  V  |  ( ( # `  t )  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E
) }  |->  ( lastS  `  x
) ) : {
w  e. Word  V  | 
( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) } -1-1-onto-> { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E } )
22 eqid 2454 . . . . . . 7  |-  { t  e. Word  V  |  ( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) }  =  {
t  e. Word  V  | 
( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) }
2322wwlkextwrd 24933 . . . . . 6  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  { t  e. Word  V  |  ( (
# `  t )  =  ( N  + 
2 )  /\  (
t substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) }  =  {
t  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( t substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) } )
2423eqcomd 2462 . . . . 5  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  { t  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  |  ( ( t substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E
) }  =  {
t  e. Word  V  | 
( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) } )
2524mpteq1d 4520 . . . 4  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( x  e.  { t  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( t substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E ) }  |->  ( lastS  `  x ) )  =  ( x  e.  {
t  e. Word  V  | 
( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) }  |->  ( lastS  `  x
) ) )
266wwlkextwrd 24933 . . . . 5  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  { w  e. Word  V  |  ( (
# `  w )  =  ( N  + 
2 )  /\  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) }  =  {
w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) } )
2726eqcomd 2462 . . . 4  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  { w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) }  =  {
w  e. Word  V  | 
( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) } )
28 eqidd 2455 . . . 4  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  { n  e.  V  |  {
( lastS  `  W ) ,  n }  e.  ran  E }  =  { n  e.  V  |  {
( lastS  `  W ) ,  n }  e.  ran  E } )
2925, 27, 28f1oeq123d 5795 . . 3  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( (
x  e.  { t  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( t substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) }  |->  ( lastS  `  x
) ) : {
w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) } -1-1-onto-> { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E }  <->  ( x  e.  { t  e. Word  V  |  ( ( # `  t )  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E
) }  |->  ( lastS  `  x
) ) : {
w  e. Word  V  | 
( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) } -1-1-onto-> { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E } ) )
3021, 29mpbird 232 . 2  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( x  e.  { t  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( t substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E ) }  |->  ( lastS  `  x ) ) : { w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) } -1-1-onto-> { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E } )
31 f1oeq1 5789 . . 3  |-  ( f  =  ( x  e. 
{ t  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( t substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E ) }  |->  ( lastS  `  x ) )  -> 
( f : {
w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) } -1-1-onto-> { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E }  <->  ( x  e.  { t  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( t substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E ) }  |->  ( lastS  `  x ) ) : { w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) } -1-1-onto-> { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E } ) )
3231spcegv 3192 . 2  |-  ( ( x  e.  { t  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( t substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) }  |->  ( lastS  `  x
) )  e.  _V  ->  ( ( x  e. 
{ t  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( t substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E ) }  |->  ( lastS  `  x ) ) : { w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) } -1-1-onto-> { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E }  ->  E. f  f : {
w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) } -1-1-onto-> { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E } ) )
335, 30, 32sylc 60 1  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  E. f 
f : { w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) } -1-1-onto-> { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E } )
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   {cpr 4018   <.cop 4022    |-> cmpt 4497   ran crn 4989   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482    + caddc 9484   2c2 10581   #chash 12390  Word cword 12521   lastS clsw 12522   substr csubstr 12525   WWalksN cwwlkn 24883
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-rp 11222  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529  df-lsw 12530  df-concat 12531  df-s1 12532  df-substr 12533  df-wwlk 24884  df-wwlkn 24885
This theorem is referenced by:  wwlkexthasheq  24939
  Copyright terms: Public domain W3C validator