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Theorem wlknwwlkninj 30369
Description: Lemma 2 for wlknwwlknbij2 30372. (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
wlknwwlkninj  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  F : T -1-1-> W )
Distinct variable groups:    E, p    N, p, t    t, T    V, p    t, W
Allowed substitution hints:    T( p)    E( t)    F( t, p)    V( t)    W( p)

Proof of Theorem wlknwwlkninj
Dummy variables  v  w 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 30368 . . 3  |-  ( N  e.  NN0  ->  F : T
--> W )
54adantl 466 . 2  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  F : T --> W )
6 fveq2 5712 . . . . . . 7  |-  ( t  =  v  ->  ( 2nd `  t )  =  ( 2nd `  v
) )
7 fvex 5722 . . . . . . 7  |-  ( 2nd `  v )  e.  _V
86, 3, 7fvmpt 5795 . . . . . 6  |-  ( v  e.  T  ->  ( F `  v )  =  ( 2nd `  v
) )
9 fveq2 5712 . . . . . . 7  |-  ( t  =  w  ->  ( 2nd `  t )  =  ( 2nd `  w
) )
10 fvex 5722 . . . . . . 7  |-  ( 2nd `  w )  e.  _V
119, 3, 10fvmpt 5795 . . . . . 6  |-  ( w  e.  T  ->  ( F `  w )  =  ( 2nd `  w
) )
128, 11eqeqan12d 2458 . . . . 5  |-  ( ( v  e.  T  /\  w  e.  T )  ->  ( ( F `  v )  =  ( F `  w )  <-> 
( 2nd `  v
)  =  ( 2nd `  w ) ) )
1312adantl 466 . . . 4  |-  ( ( ( V USGrph  E  /\  N  e.  NN0 )  /\  ( v  e.  T  /\  w  e.  T
) )  ->  (
( F `  v
)  =  ( F `
 w )  <->  ( 2nd `  v )  =  ( 2nd `  w ) ) )
14 fveq2 5712 . . . . . . . . 9  |-  ( p  =  v  ->  ( 1st `  p )  =  ( 1st `  v
) )
1514fveq2d 5716 . . . . . . . 8  |-  ( p  =  v  ->  ( # `
 ( 1st `  p
) )  =  (
# `  ( 1st `  v ) ) )
1615eqeq1d 2451 . . . . . . 7  |-  ( p  =  v  ->  (
( # `  ( 1st `  p ) )  =  N  <->  ( # `  ( 1st `  v ) )  =  N ) )
1716, 1elrab2 3140 . . . . . 6  |-  ( v  e.  T  <->  ( v  e.  ( V Walks  E )  /\  ( # `  ( 1st `  v ) )  =  N ) )
18 fveq2 5712 . . . . . . . . 9  |-  ( p  =  w  ->  ( 1st `  p )  =  ( 1st `  w
) )
1918fveq2d 5716 . . . . . . . 8  |-  ( p  =  w  ->  ( # `
 ( 1st `  p
) )  =  (
# `  ( 1st `  w ) ) )
2019eqeq1d 2451 . . . . . . 7  |-  ( p  =  w  ->  (
( # `  ( 1st `  p ) )  =  N  <->  ( # `  ( 1st `  w ) )  =  N ) )
2120, 1elrab2 3140 . . . . . 6  |-  ( w  e.  T  <->  ( w  e.  ( V Walks  E )  /\  ( # `  ( 1st `  w ) )  =  N ) )
2217, 21anbi12i 697 . . . . 5  |-  ( ( v  e.  T  /\  w  e.  T )  <->  ( ( v  e.  ( V Walks  E )  /\  ( # `  ( 1st `  v ) )  =  N )  /\  (
w  e.  ( V Walks 
E )  /\  ( # `
 ( 1st `  w
) )  =  N ) ) )
23 usg2wlkeq2 30367 . . . . . 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 ) )
24233expb 1188 . . . . 5  |-  ( ( ( 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 ) )
2522, 24sylan2b 475 . . . 4  |-  ( ( ( V USGrph  E  /\  N  e.  NN0 )  /\  ( v  e.  T  /\  w  e.  T
) )  ->  (
( 2nd `  v
)  =  ( 2nd `  w )  ->  v  =  w ) )
2613, 25sylbid 215 . . 3  |-  ( ( ( V USGrph  E  /\  N  e.  NN0 )  /\  ( v  e.  T  /\  w  e.  T
) )  ->  (
( F `  v
)  =  ( F `
 w )  -> 
v  =  w ) )
2726ralrimivva 2829 . 2  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  A. v  e.  T  A. w  e.  T  ( ( F `  v )  =  ( F `  w )  ->  v  =  w ) )
28 dff13 5992 . 2  |-  ( F : T -1-1-> W  <->  ( F : T --> W  /\  A. v  e.  T  A. w  e.  T  (
( F `  v
)  =  ( F `
 w )  -> 
v  =  w ) ) )
295, 27, 28sylanbrc 664 1  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  F : T -1-1-> W )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   {crab 2740   class class class wbr 4313    e. cmpt 4371   -->wf 5435   -1-1->wf1 5436   ` cfv 5439  (class class class)co 6112   1stc1st 6596   2ndc2nd 6597   NN0cn0 10600   #chash 12124   USGrph cusg 23286   Walks cwalk 23427   WWalksN cwwlkn 30338
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-fzo 11570  df-hash 12125  df-word 12250  df-usgra 23288  df-wlk 23437  df-wwlk 30339  df-wwlkn 30340
This theorem is referenced by:  wlknwwlknbij  30371
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