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Theorem wlkiswwlkinj 30348
Description: Lemma 2 for wlkiswwlkbij2 30351. (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 30347 . . 3  |-  ( ( P  e.  V  /\  N  e.  NN0 )  ->  F : T --> W )
543adant1 1006 . 2  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  F : T --> W )
6 fveq2 5690 . . . . . . 7  |-  ( t  =  v  ->  ( 2nd `  t )  =  ( 2nd `  v
) )
7 fvex 5700 . . . . . . 7  |-  ( 2nd `  v )  e.  _V
86, 3, 7fvmpt 5773 . . . . . 6  |-  ( v  e.  T  ->  ( F `  v )  =  ( 2nd `  v
) )
9 fveq2 5690 . . . . . . 7  |-  ( t  =  w  ->  ( 2nd `  t )  =  ( 2nd `  w
) )
10 fvex 5700 . . . . . . 7  |-  ( 2nd `  w )  e.  _V
119, 3, 10fvmpt 5773 . . . . . 6  |-  ( w  e.  T  ->  ( F `  w )  =  ( 2nd `  w
) )
128, 11eqeqan12d 2457 . . . . 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 5690 . . . . . . . . . 10  |-  ( p  =  v  ->  ( 1st `  p )  =  ( 1st `  v
) )
1514fveq2d 5694 . . . . . . . . 9  |-  ( p  =  v  ->  ( # `
 ( 1st `  p
) )  =  (
# `  ( 1st `  v ) ) )
1615eqeq1d 2450 . . . . . . . 8  |-  ( p  =  v  ->  (
( # `  ( 1st `  p ) )  =  N  <->  ( # `  ( 1st `  v ) )  =  N ) )
17 fveq2 5690 . . . . . . . . . 10  |-  ( p  =  v  ->  ( 2nd `  p )  =  ( 2nd `  v
) )
1817fveq1d 5692 . . . . . . . . 9  |-  ( p  =  v  ->  (
( 2nd `  p
) `  0 )  =  ( ( 2nd `  v ) `  0
) )
1918eqeq1d 2450 . . . . . . . 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 3118 . . . . . 6  |-  ( v  e.  T  <->  ( v  e.  ( V Walks  E )  /\  ( ( # `  ( 1st `  v
) )  =  N  /\  ( ( 2nd `  v ) `  0
)  =  P ) ) )
22 fveq2 5690 . . . . . . . . . 10  |-  ( p  =  w  ->  ( 1st `  p )  =  ( 1st `  w
) )
2322fveq2d 5694 . . . . . . . . 9  |-  ( p  =  w  ->  ( # `
 ( 1st `  p
) )  =  (
# `  ( 1st `  w ) ) )
2423eqeq1d 2450 . . . . . . . 8  |-  ( p  =  w  ->  (
( # `  ( 1st `  p ) )  =  N  <->  ( # `  ( 1st `  w ) )  =  N ) )
25 fveq2 5690 . . . . . . . . . 10  |-  ( p  =  w  ->  ( 2nd `  p )  =  ( 2nd `  w
) )
2625fveq1d 5692 . . . . . . . . 9  |-  ( p  =  w  ->  (
( 2nd `  p
) `  0 )  =  ( ( 2nd `  w ) `  0
) )
2726eqeq1d 2450 . . . . . . . 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 3118 . . . . . 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 986 . . . . . . 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 30339 . . . . . 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 1218 . . . . 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 2807 . 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 5970 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2714   {crab 2718   class class class wbr 4291    e. cmpt 4349   -->wf 5413   -1-1->wf1 5414   ` cfv 5417  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   0cc0 9281   NN0cn0 10578   #chash 12102   USGrph cusg 23263   Walks cwalk 23404   WWalksN cwwlkn 30310
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-card 8108  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-fzo 11548  df-hash 12103  df-word 12228  df-usgra 23265  df-wlk 23414  df-wwlk 30311  df-wwlkn 30312
This theorem is referenced by:  wlkiswwlkbij  30350
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