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Theorem wlknwwlknvbij 24402
Description: There is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length and starting at the same vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
Assertion
Ref Expression
wlknwwlknvbij  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  E. f 
f : { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  X ) } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  X } )
Distinct variable groups:    f, E, p, w    f, N, p, w    f, V, p, w    f, X, p, w

Proof of Theorem wlknwwlknvbij
Dummy variables  q 
s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6300 . . . . . 6  |-  ( V Walks 
E )  e.  _V
21rabex 4591 . . . . 5  |-  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  e.  _V
32mptex 6122 . . . 4  |-  ( p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  |->  ( 2nd `  p ) )  e.  _V
43resex 5308 . . 3  |-  ( ( p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  |->  ( 2nd `  p
) )  |`  { p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  | 
( ( 2nd `  p
) `  0 )  =  X } )  e. 
_V
5 eqid 2460 . . . 4  |-  ( p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  |->  ( 2nd `  p ) )  =  ( p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  |->  ( 2nd `  p ) )
6 fveq2 5857 . . . . . . . . 9  |-  ( q  =  t  ->  ( 1st `  q )  =  ( 1st `  t
) )
76fveq2d 5861 . . . . . . . 8  |-  ( q  =  t  ->  ( # `
 ( 1st `  q
) )  =  (
# `  ( 1st `  t ) ) )
87eqeq1d 2462 . . . . . . 7  |-  ( q  =  t  ->  (
( # `  ( 1st `  q ) )  =  N  <->  ( # `  ( 1st `  t ) )  =  N ) )
98cbvrabv 3105 . . . . . 6  |-  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  =  { t  e.  ( V Walks  E
)  |  ( # `  ( 1st `  t
) )  =  N }
10 eqid 2460 . . . . . 6  |-  ( ( V WWalksN  E ) `  N
)  =  ( ( V WWalksN  E ) `  N
)
11 fveq2 5857 . . . . . . 7  |-  ( p  =  s  ->  ( 2nd `  p )  =  ( 2nd `  s
) )
1211cbvmptv 4531 . . . . . 6  |-  ( p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  |->  ( 2nd `  p ) )  =  ( s  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  |->  ( 2nd `  s ) )
139, 10, 12wlknwwlknbij 24375 . . . . 5  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  (
p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  |->  ( 2nd `  p
) ) : {
q  e.  ( V Walks 
E )  |  (
# `  ( 1st `  q ) )  =  N } -1-1-onto-> ( ( V WWalksN  E
) `  N )
)
14133adant3 1011 . . . 4  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  (
p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  |->  ( 2nd `  p
) ) : {
q  e.  ( V Walks 
E )  |  (
# `  ( 1st `  q ) )  =  N } -1-1-onto-> ( ( V WWalksN  E
) `  N )
)
15 fveq1 5856 . . . . . 6  |-  ( w  =  ( 2nd `  p
)  ->  ( w `  0 )  =  ( ( 2nd `  p
) `  0 )
)
1615eqeq1d 2462 . . . . 5  |-  ( w  =  ( 2nd `  p
)  ->  ( (
w `  0 )  =  X  <->  ( ( 2nd `  p ) `  0
)  =  X ) )
17163ad2ant3 1014 . . . 4  |-  ( ( ( V USGrph  E  /\  N  e.  NN0  /\  X  e.  V )  /\  p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  /\  w  =  ( 2nd `  p ) )  -> 
( ( w ` 
0 )  =  X  <-> 
( ( 2nd `  p
) `  0 )  =  X ) )
185, 14, 17f1oresrab 6044 . . 3  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  (
( p  e.  {
q  e.  ( V Walks 
E )  |  (
# `  ( 1st `  q ) )  =  N }  |->  ( 2nd `  p ) )  |`  { p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  |  ( ( 2nd `  p ) `
 0 )  =  X } ) : { p  e.  {
q  e.  ( V Walks 
E )  |  (
# `  ( 1st `  q ) )  =  N }  |  ( ( 2nd `  p
) `  0 )  =  X } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  X } )
19 f1oeq1 5798 . . . 4  |-  ( f  =  ( ( p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  |->  ( 2nd `  p ) )  |`  { p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  | 
( ( 2nd `  p
) `  0 )  =  X } )  -> 
( f : {
p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  |  ( ( 2nd `  p ) `
 0 )  =  X } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  X }  <->  ( (
p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  |->  ( 2nd `  p
) )  |`  { p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  | 
( ( 2nd `  p
) `  0 )  =  X } ) : { p  e.  {
q  e.  ( V Walks 
E )  |  (
# `  ( 1st `  q ) )  =  N }  |  ( ( 2nd `  p
) `  0 )  =  X } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  X } ) )
2019spcegv 3192 . . 3  |-  ( ( ( p  e.  {
q  e.  ( V Walks 
E )  |  (
# `  ( 1st `  q ) )  =  N }  |->  ( 2nd `  p ) )  |`  { p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  |  ( ( 2nd `  p ) `
 0 )  =  X } )  e. 
_V  ->  ( ( ( p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  |->  ( 2nd `  p
) )  |`  { p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  | 
( ( 2nd `  p
) `  0 )  =  X } ) : { p  e.  {
q  e.  ( V Walks 
E )  |  (
# `  ( 1st `  q ) )  =  N }  |  ( ( 2nd `  p
) `  0 )  =  X } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  X }  ->  E. f  f : {
p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  |  ( ( 2nd `  p ) `
 0 )  =  X } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  X } ) )
214, 18, 20mpsyl 63 . 2  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  E. f 
f : { p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  | 
( ( 2nd `  p
) `  0 )  =  X } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  X } )
22 df-rab 2816 . . . . 5  |-  { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  X ) }  =  { p  |  ( p  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  X ) ) }
23 anass 649 . . . . . . 7  |-  ( ( ( p  e.  ( V Walks  E )  /\  ( # `  ( 1st `  p ) )  =  N )  /\  (
( 2nd `  p
) `  0 )  =  X )  <->  ( p  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  X ) ) )
2423bicomi 202 . . . . . 6  |-  ( ( p  e.  ( V Walks 
E )  /\  (
( # `  ( 1st `  p ) )  =  N  /\  ( ( 2nd `  p ) `
 0 )  =  X ) )  <->  ( (
p  e.  ( V Walks 
E )  /\  ( # `
 ( 1st `  p
) )  =  N )  /\  ( ( 2nd `  p ) `
 0 )  =  X ) )
2524abbii 2594 . . . . 5  |-  { p  |  ( p  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  X ) ) }  =  {
p  |  ( ( p  e.  ( V Walks 
E )  /\  ( # `
 ( 1st `  p
) )  =  N )  /\  ( ( 2nd `  p ) `
 0 )  =  X ) }
26 fveq2 5857 . . . . . . . . . . . 12  |-  ( q  =  p  ->  ( 1st `  q )  =  ( 1st `  p
) )
2726fveq2d 5861 . . . . . . . . . . 11  |-  ( q  =  p  ->  ( # `
 ( 1st `  q
) )  =  (
# `  ( 1st `  p ) ) )
2827eqeq1d 2462 . . . . . . . . . 10  |-  ( q  =  p  ->  (
( # `  ( 1st `  q ) )  =  N  <->  ( # `  ( 1st `  p ) )  =  N ) )
2928elrab 3254 . . . . . . . . 9  |-  ( p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  <->  ( p  e.  ( V Walks  E )  /\  ( # `  ( 1st `  p ) )  =  N ) )
3029anbi1i 695 . . . . . . . 8  |-  ( ( p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  /\  ( ( 2nd `  p ) `
 0 )  =  X )  <->  ( (
p  e.  ( V Walks 
E )  /\  ( # `
 ( 1st `  p
) )  =  N )  /\  ( ( 2nd `  p ) `
 0 )  =  X ) )
3130bicomi 202 . . . . . . 7  |-  ( ( ( p  e.  ( V Walks  E )  /\  ( # `  ( 1st `  p ) )  =  N )  /\  (
( 2nd `  p
) `  0 )  =  X )  <->  ( p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  /\  ( ( 2nd `  p
) `  0 )  =  X ) )
3231abbii 2594 . . . . . 6  |-  { p  |  ( ( p  e.  ( V Walks  E
)  /\  ( # `  ( 1st `  p ) )  =  N )  /\  ( ( 2nd `  p
) `  0 )  =  X ) }  =  { p  |  (
p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  /\  ( ( 2nd `  p ) `
 0 )  =  X ) }
33 df-rab 2816 . . . . . 6  |-  { p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  | 
( ( 2nd `  p
) `  0 )  =  X }  =  {
p  |  ( p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  /\  ( ( 2nd `  p
) `  0 )  =  X ) }
3432, 33eqtr4i 2492 . . . . 5  |-  { p  |  ( ( p  e.  ( V Walks  E
)  /\  ( # `  ( 1st `  p ) )  =  N )  /\  ( ( 2nd `  p
) `  0 )  =  X ) }  =  { p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  |  ( ( 2nd `  p ) `
 0 )  =  X }
3522, 25, 343eqtri 2493 . . . 4  |-  { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  X ) }  =  { p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  | 
( ( 2nd `  p
) `  0 )  =  X }
36 f1oeq2 5799 . . . 4  |-  ( { p  e.  ( V Walks 
E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  X ) }  =  { p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  |  ( ( 2nd `  p ) `
 0 )  =  X }  ->  (
f : { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  X ) } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  X }  <->  f : { p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  |  ( ( 2nd `  p ) `
 0 )  =  X } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  X } ) )
3735, 36mp1i 12 . . 3  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  (
f : { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  X ) } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  X }  <->  f : { p  e.  { q  e.  ( V Walks  E
)  |  ( # `  ( 1st `  q
) )  =  N }  |  ( ( 2nd `  p ) `
 0 )  =  X } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  X } ) )
3837exbidv 1685 . 2  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( E. f  f : { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  X ) } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  X }  <->  E. f 
f : { p  e.  { q  e.  ( V Walks  E )  |  ( # `  ( 1st `  q ) )  =  N }  | 
( ( 2nd `  p
) `  0 )  =  X } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  X } ) )
3921, 38mpbird 232 1  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  E. f 
f : { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  X ) } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  X } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762   {cab 2445   {crab 2811   _Vcvv 3106   class class class wbr 4440    |-> cmpt 4498    |` cres 4994   -1-1-onto->wf1o 5578   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   0cc0 9481   NN0cn0 10784   #chash 12360   USGrph cusg 23993   Walks cwalk 24160   WWalksN cwwlkn 24340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-usgra 23996  df-wlk 24170  df-wwlk 24341  df-wwlkn 24342
This theorem is referenced by:  rusgranumwwlkg  24621
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