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Theorem wlkiswwlkinj 24844
Description: Lemma 2 for wlkiswwlkbij2 24847. (Contributed by Alexander van der Vekens, 23-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.)
Hypotheses
Ref Expression
wlkiswwlkbij.t  |-  T  =  { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) }
wlkiswwlkbij.w  |-  W  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P }
wlkiswwlkbij.f  |-  F  =  ( t  e.  T  |->  ( 2nd `  t
) )
Assertion
Ref Expression
wlkiswwlkinj  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  F : T -1-1-> W )
Distinct variable groups:    E, p, t, w    N, p, t, w    P, p, t, w   
t, T    V, p, t, w    t, W    w, F    w, T
Allowed substitution hints:    T( p)    F( t, p)    W( w, p)

Proof of Theorem wlkiswwlkinj
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 wlkiswwlkbij.t . . . 4  |-  T  =  { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) }
2 wlkiswwlkbij.w . . . 4  |-  W  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P }
3 wlkiswwlkbij.f . . . 4  |-  F  =  ( t  e.  T  |->  ( 2nd `  t
) )
41, 2, 3wlkiswwlkfun 24843 . . 3  |-  ( ( P  e.  V  /\  N  e.  NN0 )  ->  F : T --> W )
543adant1 1014 . 2  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  F : T --> W )
6 fveq2 5872 . . . . . . 7  |-  ( t  =  v  ->  ( 2nd `  t )  =  ( 2nd `  v
) )
7 fvex 5882 . . . . . . 7  |-  ( 2nd `  v )  e.  _V
86, 3, 7fvmpt 5956 . . . . . 6  |-  ( v  e.  T  ->  ( F `  v )  =  ( 2nd `  v
) )
9 fveq2 5872 . . . . . . 7  |-  ( t  =  w  ->  ( 2nd `  t )  =  ( 2nd `  w
) )
10 fvex 5882 . . . . . . 7  |-  ( 2nd `  w )  e.  _V
119, 3, 10fvmpt 5956 . . . . . 6  |-  ( w  e.  T  ->  ( F `  w )  =  ( 2nd `  w
) )
128, 11eqeqan12d 2480 . . . . 5  |-  ( ( v  e.  T  /\  w  e.  T )  ->  ( ( F `  v )  =  ( F `  w )  <-> 
( 2nd `  v
)  =  ( 2nd `  w ) ) )
1312adantl 466 . . . 4  |-  ( ( ( V USGrph  E  /\  P  e.  V  /\  N  e.  NN0 )  /\  ( v  e.  T  /\  w  e.  T
) )  ->  (
( F `  v
)  =  ( F `
 w )  <->  ( 2nd `  v )  =  ( 2nd `  w ) ) )
14 fveq2 5872 . . . . . . . . . 10  |-  ( p  =  v  ->  ( 1st `  p )  =  ( 1st `  v
) )
1514fveq2d 5876 . . . . . . . . 9  |-  ( p  =  v  ->  ( # `
 ( 1st `  p
) )  =  (
# `  ( 1st `  v ) ) )
1615eqeq1d 2459 . . . . . . . 8  |-  ( p  =  v  ->  (
( # `  ( 1st `  p ) )  =  N  <->  ( # `  ( 1st `  v ) )  =  N ) )
17 fveq2 5872 . . . . . . . . . 10  |-  ( p  =  v  ->  ( 2nd `  p )  =  ( 2nd `  v
) )
1817fveq1d 5874 . . . . . . . . 9  |-  ( p  =  v  ->  (
( 2nd `  p
) `  0 )  =  ( ( 2nd `  v ) `  0
) )
1918eqeq1d 2459 . . . . . . . 8  |-  ( p  =  v  ->  (
( ( 2nd `  p
) `  0 )  =  P  <->  ( ( 2nd `  v ) `  0
)  =  P ) )
2016, 19anbi12d 710 . . . . . . 7  |-  ( p  =  v  ->  (
( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P )  <->  ( ( # `
 ( 1st `  v
) )  =  N  /\  ( ( 2nd `  v ) `  0
)  =  P ) ) )
2120, 1elrab2 3259 . . . . . 6  |-  ( v  e.  T  <->  ( v  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  v
) )  =  N  /\  ( ( 2nd `  v ) `  0
)  =  P ) ) )
22 fveq2 5872 . . . . . . . . . 10  |-  ( p  =  w  ->  ( 1st `  p )  =  ( 1st `  w
) )
2322fveq2d 5876 . . . . . . . . 9  |-  ( p  =  w  ->  ( # `
 ( 1st `  p
) )  =  (
# `  ( 1st `  w ) ) )
2423eqeq1d 2459 . . . . . . . 8  |-  ( p  =  w  ->  (
( # `  ( 1st `  p ) )  =  N  <->  ( # `  ( 1st `  w ) )  =  N ) )
25 fveq2 5872 . . . . . . . . . 10  |-  ( p  =  w  ->  ( 2nd `  p )  =  ( 2nd `  w
) )
2625fveq1d 5874 . . . . . . . . 9  |-  ( p  =  w  ->  (
( 2nd `  p
) `  0 )  =  ( ( 2nd `  w ) `  0
) )
2726eqeq1d 2459 . . . . . . . 8  |-  ( p  =  w  ->  (
( ( 2nd `  p
) `  0 )  =  P  <->  ( ( 2nd `  w ) `  0
)  =  P ) )
2824, 27anbi12d 710 . . . . . . 7  |-  ( p  =  w  ->  (
( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P )  <->  ( ( # `
 ( 1st `  w
) )  =  N  /\  ( ( 2nd `  w ) `  0
)  =  P ) ) )
2928, 1elrab2 3259 . . . . . 6  |-  ( w  e.  T  <->  ( w  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  w
) )  =  N  /\  ( ( 2nd `  w ) `  0
)  =  P ) ) )
3021, 29anbi12i 697 . . . . 5  |-  ( ( v  e.  T  /\  w  e.  T )  <->  ( ( v  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  v ) )  =  N  /\  (
( 2nd `  v
) `  0 )  =  P ) )  /\  ( w  e.  ( V Walks  E )  /\  (
( # `  ( 1st `  w ) )  =  N  /\  ( ( 2nd `  w ) `
 0 )  =  P ) ) ) )
31 3simpb 994 . . . . . . 7  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  ( V USGrph  E  /\  N  e. 
NN0 ) )
3231adantr 465 . . . . . 6  |-  ( ( ( V USGrph  E  /\  P  e.  V  /\  N  e.  NN0 )  /\  ( ( v  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  v
) )  =  N  /\  ( ( 2nd `  v ) `  0
)  =  P ) )  /\  ( w  e.  ( V Walks  E
)  /\  ( ( # `
 ( 1st `  w
) )  =  N  /\  ( ( 2nd `  w ) `  0
)  =  P ) ) ) )  -> 
( V USGrph  E  /\  N  e.  NN0 ) )
33 simpl 457 . . . . . . . . 9  |-  ( ( ( # `  ( 1st `  v ) )  =  N  /\  (
( 2nd `  v
) `  0 )  =  P )  ->  ( # `
 ( 1st `  v
) )  =  N )
3433anim2i 569 . . . . . . . 8  |-  ( ( v  e.  ( V Walks 
E )  /\  (
( # `  ( 1st `  v ) )  =  N  /\  ( ( 2nd `  v ) `
 0 )  =  P ) )  -> 
( v  e.  ( V Walks  E )  /\  ( # `  ( 1st `  v ) )  =  N ) )
3534adantr 465 . . . . . . 7  |-  ( ( ( v  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  v ) )  =  N  /\  (
( 2nd `  v
) `  0 )  =  P ) )  /\  ( w  e.  ( V Walks  E )  /\  (
( # `  ( 1st `  w ) )  =  N  /\  ( ( 2nd `  w ) `
 0 )  =  P ) ) )  ->  ( v  e.  ( V Walks  E )  /\  ( # `  ( 1st `  v ) )  =  N ) )
3635adantl 466 . . . . . 6  |-  ( ( ( V USGrph  E  /\  P  e.  V  /\  N  e.  NN0 )  /\  ( ( v  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  v
) )  =  N  /\  ( ( 2nd `  v ) `  0
)  =  P ) )  /\  ( w  e.  ( V Walks  E
)  /\  ( ( # `
 ( 1st `  w
) )  =  N  /\  ( ( 2nd `  w ) `  0
)  =  P ) ) ) )  -> 
( v  e.  ( V Walks  E )  /\  ( # `  ( 1st `  v ) )  =  N ) )
37 simpl 457 . . . . . . . . 9  |-  ( ( ( # `  ( 1st `  w ) )  =  N  /\  (
( 2nd `  w
) `  0 )  =  P )  ->  ( # `
 ( 1st `  w
) )  =  N )
3837anim2i 569 . . . . . . . 8  |-  ( ( w  e.  ( V Walks 
E )  /\  (
( # `  ( 1st `  w ) )  =  N  /\  ( ( 2nd `  w ) `
 0 )  =  P ) )  -> 
( w  e.  ( V Walks  E )  /\  ( # `  ( 1st `  w ) )  =  N ) )
3938adantl 466 . . . . . . 7  |-  ( ( ( v  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  v ) )  =  N  /\  (
( 2nd `  v
) `  0 )  =  P ) )  /\  ( w  e.  ( V Walks  E )  /\  (
( # `  ( 1st `  w ) )  =  N  /\  ( ( 2nd `  w ) `
 0 )  =  P ) ) )  ->  ( w  e.  ( V Walks  E )  /\  ( # `  ( 1st `  w ) )  =  N ) )
4039adantl 466 . . . . . 6  |-  ( ( ( V USGrph  E  /\  P  e.  V  /\  N  e.  NN0 )  /\  ( ( v  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  v
) )  =  N  /\  ( ( 2nd `  v ) `  0
)  =  P ) )  /\  ( w  e.  ( V Walks  E
)  /\  ( ( # `
 ( 1st `  w
) )  =  N  /\  ( ( 2nd `  w ) `  0
)  =  P ) ) ) )  -> 
( w  e.  ( V Walks  E )  /\  ( # `  ( 1st `  w ) )  =  N ) )
41 usg2wlkeq2 24835 . . . . . 6  |-  ( ( ( V USGrph  E  /\  N  e.  NN0 )  /\  ( v  e.  ( V Walks  E )  /\  ( # `  ( 1st `  v ) )  =  N )  /\  (
w  e.  ( V Walks 
E )  /\  ( # `
 ( 1st `  w
) )  =  N ) )  ->  (
( 2nd `  v
)  =  ( 2nd `  w )  ->  v  =  w ) )
4232, 36, 40, 41syl3anc 1228 . . . . 5  |-  ( ( ( V USGrph  E  /\  P  e.  V  /\  N  e.  NN0 )  /\  ( ( v  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  v
) )  =  N  /\  ( ( 2nd `  v ) `  0
)  =  P ) )  /\  ( w  e.  ( V Walks  E
)  /\  ( ( # `
 ( 1st `  w
) )  =  N  /\  ( ( 2nd `  w ) `  0
)  =  P ) ) ) )  -> 
( ( 2nd `  v
)  =  ( 2nd `  w )  ->  v  =  w ) )
4330, 42sylan2b 475 . . . 4  |-  ( ( ( V USGrph  E  /\  P  e.  V  /\  N  e.  NN0 )  /\  ( v  e.  T  /\  w  e.  T
) )  ->  (
( 2nd `  v
)  =  ( 2nd `  w )  ->  v  =  w ) )
4413, 43sylbid 215 . . 3  |-  ( ( ( V USGrph  E  /\  P  e.  V  /\  N  e.  NN0 )  /\  ( v  e.  T  /\  w  e.  T
) )  ->  (
( F `  v
)  =  ( F `
 w )  -> 
v  =  w ) )
4544ralrimivva 2878 . 2  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  A. v  e.  T  A. w  e.  T  ( ( F `  v )  =  ( F `  w )  ->  v  =  w ) )
46 dff13 6167 . 2  |-  ( F : T -1-1-> W  <->  ( F : T --> W  /\  A. v  e.  T  A. w  e.  T  (
( F `  v
)  =  ( F `
 w )  -> 
v  =  w ) ) )
475, 45, 46sylanbrc 664 1  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  F : T -1-1-> W )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   class class class wbr 4456    |-> cmpt 4515   -->wf 5590   -1-1->wf1 5591   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   0cc0 9509   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-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:  wlkiswwlkbij  24846
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