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

Theorem wlknwwlknsur 24838
Description: Lemma 3 for wlknwwlknbij2 24840. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
Hypotheses
Ref Expression
wlknwwlknbij.t  |-  T  =  { p  e.  ( V Walks  E )  |  ( # `  ( 1st `  p ) )  =  N }
wlknwwlknbij.w  |-  W  =  ( ( V WWalksN  E
) `  N )
wlknwwlknbij.f  |-  F  =  ( t  e.  T  |->  ( 2nd `  t
) )
Assertion
Ref Expression
wlknwwlknsur  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  F : T -onto-> W )
Distinct variable groups:    E, p    N, p, t    t, T    V, p    t, W    F, p    T, p    W, p
Allowed substitution hints:    E( t)    F( t)    V( t)

Proof of Theorem wlknwwlknsur
Dummy variables  f  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wlknwwlknbij.t . . . 4  |-  T  =  { p  e.  ( V Walks  E )  |  ( # `  ( 1st `  p ) )  =  N }
2 wlknwwlknbij.w . . . 4  |-  W  =  ( ( V WWalksN  E
) `  N )
3 wlknwwlknbij.f . . . 4  |-  F  =  ( t  e.  T  |->  ( 2nd `  t
) )
41, 2, 3wlknwwlknfun 24836 . . 3  |-  ( N  e.  NN0  ->  F : T
--> W )
54adantl 466 . 2  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  F : T --> W )
62eleq2i 2535 . . . . 5  |-  ( p  e.  W  <->  p  e.  ( ( V WWalksN  E
) `  N )
)
7 wlklniswwlkn 24827 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( E. f
( f ( V Walks 
E ) p  /\  ( # `  f )  =  N )  <->  p  e.  ( ( V WWalksN  E
) `  N )
) )
8 df-br 4457 . . . . . . . . . . . . 13  |-  ( f ( V Walks  E ) p  <->  <. f ,  p >.  e.  ( V Walks  E
) )
9 vex 3112 . . . . . . . . . . . . . . . . 17  |-  f  e. 
_V
10 vex 3112 . . . . . . . . . . . . . . . . 17  |-  p  e. 
_V
119, 10op1st 6807 . . . . . . . . . . . . . . . 16  |-  ( 1st `  <. f ,  p >. )  =  f
1211eqcomi 2470 . . . . . . . . . . . . . . 15  |-  f  =  ( 1st `  <. f ,  p >. )
1312fveq2i 5875 . . . . . . . . . . . . . 14  |-  ( # `  f )  =  (
# `  ( 1st ` 
<. f ,  p >. ) )
1413eqeq1i 2464 . . . . . . . . . . . . 13  |-  ( (
# `  f )  =  N  <->  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )
15 elex 3118 . . . . . . . . . . . . . . 15  |-  ( <.
f ,  p >.  e.  ( V Walks  E )  ->  <. f ,  p >.  e.  _V )
16 eleq1 2529 . . . . . . . . . . . . . . . . . . 19  |-  ( u  =  <. f ,  p >.  ->  ( u  e.  ( V Walks  E )  <->  <. f ,  p >.  e.  ( V Walks  E ) ) )
1716biimparc 487 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. f ,  p >.  e.  ( V Walks  E )  /\  u  =  <. f ,  p >. )  ->  u  e.  ( V Walks 
E ) )
1817adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( <. f ,  p >.  e.  ( V Walks  E
)  /\  u  =  <. f ,  p >. )  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  ->  u  e.  ( V Walks  E ) )
19 fveq2 5872 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  <. f ,  p >.  ->  ( 1st `  u
)  =  ( 1st `  <. f ,  p >. ) )
2019fveq2d 5876 . . . . . . . . . . . . . . . . . . . 20  |-  ( u  =  <. f ,  p >.  ->  ( # `  ( 1st `  u ) )  =  ( # `  ( 1st `  <. f ,  p >. ) ) )
2120eqeq1d 2459 . . . . . . . . . . . . . . . . . . 19  |-  ( u  =  <. f ,  p >.  ->  ( ( # `  ( 1st `  u
) )  =  N  <-> 
( # `  ( 1st `  <. f ,  p >. ) )  =  N ) )
2221adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. f ,  p >.  e.  ( V Walks  E )  /\  u  =  <. f ,  p >. )  ->  ( ( # `  ( 1st `  u ) )  =  N  <->  ( # `  ( 1st `  <. f ,  p >. ) )  =  N ) )
2322biimpar 485 . . . . . . . . . . . . . . . . 17  |-  ( ( ( <. f ,  p >.  e.  ( V Walks  E
)  /\  u  =  <. f ,  p >. )  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  ->  ( # `  ( 1st `  u ) )  =  N )
24 fveq2 5872 . . . . . . . . . . . . . . . . . . . 20  |-  ( u  =  <. f ,  p >.  ->  ( 2nd `  u
)  =  ( 2nd `  <. f ,  p >. ) )
259, 10op2nd 6808 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2nd `  <. f ,  p >. )  =  p
2624, 25syl6req 2515 . . . . . . . . . . . . . . . . . . 19  |-  ( u  =  <. f ,  p >.  ->  p  =  ( 2nd `  u ) )
2726adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. f ,  p >.  e.  ( V Walks  E )  /\  u  =  <. f ,  p >. )  ->  p  =  ( 2nd `  u ) )
2827adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( <. f ,  p >.  e.  ( V Walks  E
)  /\  u  =  <. f ,  p >. )  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  ->  p  =  ( 2nd `  u ) )
2918, 23, 28jca31 534 . . . . . . . . . . . . . . . 16  |-  ( ( ( <. f ,  p >.  e.  ( V Walks  E
)  /\  u  =  <. f ,  p >. )  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  ->  ( (
u  e.  ( V Walks 
E )  /\  ( # `
 ( 1st `  u
) )  =  N )  /\  p  =  ( 2nd `  u
) ) )
3029ex 434 . . . . . . . . . . . . . . 15  |-  ( (
<. f ,  p >.  e.  ( V Walks  E )  /\  u  =  <. f ,  p >. )  ->  ( ( # `  ( 1st `  <. f ,  p >. ) )  =  N  ->  ( ( u  e.  ( V Walks  E
)  /\  ( # `  ( 1st `  u ) )  =  N )  /\  p  =  ( 2nd `  u ) ) ) )
3115, 30spcimedv 3193 . . . . . . . . . . . . . 14  |-  ( <.
f ,  p >.  e.  ( V Walks  E )  ->  ( ( # `  ( 1st `  <. f ,  p >. )
)  =  N  ->  E. u ( ( u  e.  ( V Walks  E
)  /\  ( # `  ( 1st `  u ) )  =  N )  /\  p  =  ( 2nd `  u ) ) ) )
3231imp 429 . . . . . . . . . . . . 13  |-  ( (
<. f ,  p >.  e.  ( V Walks  E )  /\  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )  ->  E. u
( ( u  e.  ( V Walks  E )  /\  ( # `  ( 1st `  u ) )  =  N )  /\  p  =  ( 2nd `  u ) ) )
338, 14, 32syl2anb 479 . . . . . . . . . . . 12  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  N )  ->  E. u
( ( u  e.  ( V Walks  E )  /\  ( # `  ( 1st `  u ) )  =  N )  /\  p  =  ( 2nd `  u ) ) )
3433exlimiv 1723 . . . . . . . . . . 11  |-  ( E. f ( f ( V Walks  E ) p  /\  ( # `  f
)  =  N )  ->  E. u ( ( u  e.  ( V Walks 
E )  /\  ( # `
 ( 1st `  u
) )  =  N )  /\  p  =  ( 2nd `  u
) ) )
357, 34syl6bir 229 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( p  e.  ( ( V WWalksN  E
) `  N )  ->  E. u ( ( u  e.  ( V Walks 
E )  /\  ( # `
 ( 1st `  u
) )  =  N )  /\  p  =  ( 2nd `  u
) ) ) )
3635imp 429 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  p  e.  ( ( V WWalksN  E
) `  N )
)  ->  E. u
( ( u  e.  ( V Walks  E )  /\  ( # `  ( 1st `  u ) )  =  N )  /\  p  =  ( 2nd `  u ) ) )
37 fveq2 5872 . . . . . . . . . . . . . 14  |-  ( p  =  u  ->  ( 1st `  p )  =  ( 1st `  u
) )
3837fveq2d 5876 . . . . . . . . . . . . 13  |-  ( p  =  u  ->  ( # `
 ( 1st `  p
) )  =  (
# `  ( 1st `  u ) ) )
3938eqeq1d 2459 . . . . . . . . . . . 12  |-  ( p  =  u  ->  (
( # `  ( 1st `  p ) )  =  N  <->  ( # `  ( 1st `  u ) )  =  N ) )
4039elrab 3257 . . . . . . . . . . 11  |-  ( u  e.  { p  e.  ( V Walks  E )  |  ( # `  ( 1st `  p ) )  =  N }  <->  ( u  e.  ( V Walks  E )  /\  ( # `  ( 1st `  u ) )  =  N ) )
4140anbi1i 695 . . . . . . . . . 10  |-  ( ( u  e.  { p  e.  ( V Walks  E )  |  ( # `  ( 1st `  p ) )  =  N }  /\  p  =  ( 2nd `  u ) )  <->  ( (
u  e.  ( V Walks 
E )  /\  ( # `
 ( 1st `  u
) )  =  N )  /\  p  =  ( 2nd `  u
) ) )
4241exbii 1668 . . . . . . . . 9  |-  ( E. u ( u  e. 
{ p  e.  ( V Walks  E )  |  ( # `  ( 1st `  p ) )  =  N }  /\  p  =  ( 2nd `  u ) )  <->  E. u
( ( u  e.  ( V Walks  E )  /\  ( # `  ( 1st `  u ) )  =  N )  /\  p  =  ( 2nd `  u ) ) )
4336, 42sylibr 212 . . . . . . . 8  |-  ( ( V USGrph  E  /\  p  e.  ( ( V WWalksN  E
) `  N )
)  ->  E. u
( u  e.  {
p  e.  ( V Walks 
E )  |  (
# `  ( 1st `  p ) )  =  N }  /\  p  =  ( 2nd `  u
) ) )
44 df-rex 2813 . . . . . . . 8  |-  ( E. u  e.  { p  e.  ( V Walks  E )  |  ( # `  ( 1st `  p ) )  =  N } p  =  ( 2nd `  u
)  <->  E. u ( u  e.  { p  e.  ( V Walks  E )  |  ( # `  ( 1st `  p ) )  =  N }  /\  p  =  ( 2nd `  u ) ) )
4543, 44sylibr 212 . . . . . . 7  |-  ( ( V USGrph  E  /\  p  e.  ( ( V WWalksN  E
) `  N )
)  ->  E. u  e.  { p  e.  ( V Walks  E )  |  ( # `  ( 1st `  p ) )  =  N } p  =  ( 2nd `  u
) )
461rexeqi 3059 . . . . . . 7  |-  ( E. u  e.  T  p  =  ( 2nd `  u
)  <->  E. u  e.  {
p  e.  ( V Walks 
E )  |  (
# `  ( 1st `  p ) )  =  N } p  =  ( 2nd `  u
) )
4745, 46sylibr 212 . . . . . 6  |-  ( ( V USGrph  E  /\  p  e.  ( ( V WWalksN  E
) `  N )
)  ->  E. u  e.  T  p  =  ( 2nd `  u ) )
48 fveq2 5872 . . . . . . . . 9  |-  ( t  =  u  ->  ( 2nd `  t )  =  ( 2nd `  u
) )
49 fvex 5882 . . . . . . . . 9  |-  ( 2nd `  u )  e.  _V
5048, 3, 49fvmpt 5956 . . . . . . . 8  |-  ( u  e.  T  ->  ( F `  u )  =  ( 2nd `  u
) )
5150eqeq2d 2471 . . . . . . 7  |-  ( u  e.  T  ->  (
p  =  ( F `
 u )  <->  p  =  ( 2nd `  u ) ) )
5251rexbiia 2958 . . . . . 6  |-  ( E. u  e.  T  p  =  ( F `  u )  <->  E. u  e.  T  p  =  ( 2nd `  u ) )
5347, 52sylibr 212 . . . . 5  |-  ( ( V USGrph  E  /\  p  e.  ( ( V WWalksN  E
) `  N )
)  ->  E. u  e.  T  p  =  ( F `  u ) )
546, 53sylan2b 475 . . . 4  |-  ( ( V USGrph  E  /\  p  e.  W )  ->  E. u  e.  T  p  =  ( F `  u ) )
5554ralrimiva 2871 . . 3  |-  ( V USGrph  E  ->  A. p  e.  W  E. u  e.  T  p  =  ( F `  u ) )
5655adantr 465 . 2  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  A. p  e.  W  E. u  e.  T  p  =  ( F `  u ) )
57 dffo3 6047 . 2  |-  ( F : T -onto-> W  <->  ( F : T --> W  /\  A. p  e.  W  E. u  e.  T  p  =  ( F `  u ) ) )
585, 56, 57sylanbrc 664 1  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  F : T -onto-> W )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109   <.cop 4038   class class class wbr 4456    |-> cmpt 4515   -->wf 5590   -onto->wfo 5592   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   NN0cn0 10816   #chash 12407   USGrph cusg 24456   Walks cwalk 24624   WWalksN cwwlkn 24804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11821  df-hash 12408  df-word 12545  df-usgra 24459  df-wlk 24634  df-wwlk 24805  df-wwlkn 24806
This theorem is referenced by:  wlknwwlknbij  24839
  Copyright terms: Public domain W3C validator