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Theorem wlknwwlknsur 30344
Description: Lemma 3 for wlknwwlknbij2 30346. (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 30342 . . 3  |-  ( N  e.  NN0  ->  F : T
--> W )
54adantl 466 . 2  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  F : T --> W )
62eleq2i 2507 . . . . 5  |-  ( p  e.  W  <->  p  e.  ( ( V WWalksN  E
) `  N )
)
7 wlklniswwlkn 30335 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( E. f
( f ( V Walks 
E ) p  /\  ( # `  f )  =  N )  <->  p  e.  ( ( V WWalksN  E
) `  N )
) )
8 df-br 4293 . . . . . . . . . . . . 13  |-  ( f ( V Walks  E ) p  <->  <. f ,  p >.  e.  ( V Walks  E
) )
9 vex 2975 . . . . . . . . . . . . . . . . 17  |-  f  e. 
_V
10 vex 2975 . . . . . . . . . . . . . . . . 17  |-  p  e. 
_V
119, 10op1st 6585 . . . . . . . . . . . . . . . 16  |-  ( 1st `  <. f ,  p >. )  =  f
1211eqcomi 2447 . . . . . . . . . . . . . . 15  |-  f  =  ( 1st `  <. f ,  p >. )
1312fveq2i 5694 . . . . . . . . . . . . . 14  |-  ( # `  f )  =  (
# `  ( 1st ` 
<. f ,  p >. ) )
1413eqeq1i 2450 . . . . . . . . . . . . 13  |-  ( (
# `  f )  =  N  <->  ( # `  ( 1st `  <. f ,  p >. ) )  =  N )
15 elex 2981 . . . . . . . . . . . . . . 15  |-  ( <.
f ,  p >.  e.  ( V Walks  E )  ->  <. f ,  p >.  e.  _V )
16 eleq1 2503 . . . . . . . . . . . . . . . . . . 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 5691 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  <. f ,  p >.  ->  ( 1st `  u
)  =  ( 1st `  <. f ,  p >. ) )
2019fveq2d 5695 . . . . . . . . . . . . . . . . . . . 20  |-  ( u  =  <. f ,  p >.  ->  ( # `  ( 1st `  u ) )  =  ( # `  ( 1st `  <. f ,  p >. ) ) )
2120eqeq1d 2451 . . . . . . . . . . . . . . . . . . 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 5691 . . . . . . . . . . . . . . . . . . . 20  |-  ( u  =  <. f ,  p >.  ->  ( 2nd `  u
)  =  ( 2nd `  <. f ,  p >. ) )
259, 10op2nd 6586 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2nd `  <. f ,  p >. )  =  p
2624, 25syl6req 2492 . . . . . . . . . . . . . . . . . . 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 3056 . . . . . . . . . . . . . 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 1688 . . . . . . . . . . 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 5691 . . . . . . . . . . . . . 14  |-  ( p  =  u  ->  ( 1st `  p )  =  ( 1st `  u
) )
3837fveq2d 5695 . . . . . . . . . . . . 13  |-  ( p  =  u  ->  ( # `
 ( 1st `  p
) )  =  (
# `  ( 1st `  u ) ) )
3938eqeq1d 2451 . . . . . . . . . . . 12  |-  ( p  =  u  ->  (
( # `  ( 1st `  p ) )  =  N  <->  ( # `  ( 1st `  u ) )  =  N ) )
4039elrab 3117 . . . . . . . . . . 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 1634 . . . . . . . . 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 2721 . . . . . . . 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 2922 . . . . . . 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 5691 . . . . . . . . 9  |-  ( t  =  u  ->  ( 2nd `  t )  =  ( 2nd `  u
) )
49 fvex 5701 . . . . . . . . 9  |-  ( 2nd `  u )  e.  _V
5048, 3, 49fvmpt 5774 . . . . . . . 8  |-  ( u  e.  T  ->  ( F `  u )  =  ( 2nd `  u
) )
5150eqeq2d 2454 . . . . . . 7  |-  ( u  e.  T  ->  (
p  =  ( F `
 u )  <->  p  =  ( 2nd `  u ) ) )
5251rexbiia 2748 . . . . . 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 2799 . . 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 5858 . 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 1369   E.wex 1586    e. wcel 1756   A.wral 2715   E.wrex 2716   {crab 2719   _Vcvv 2972   <.cop 3883   class class class wbr 4292    e. cmpt 4350   -->wf 5414   -onto->wfo 5416   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   NN0cn0 10579   #chash 12103   USGrph cusg 23264   Walks cwalk 23405   WWalksN cwwlkn 30312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-usgra 23266  df-wlk 23415  df-wwlk 30313  df-wwlkn 30314
This theorem is referenced by:  wlknwwlknbij  30345
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