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Theorem wlknwwlkninj 24916
Description: Lemma 2 for wlknwwlknbij2 24919. (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 24915 . . 3  |-  ( N  e.  NN0  ->  F : T
--> W )
54adantl 464 . 2  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  F : T --> W )
6 fveq2 5848 . . . . . . 7  |-  ( t  =  v  ->  ( 2nd `  t )  =  ( 2nd `  v
) )
7 fvex 5858 . . . . . . 7  |-  ( 2nd `  v )  e.  _V
86, 3, 7fvmpt 5931 . . . . . 6  |-  ( v  e.  T  ->  ( F `  v )  =  ( 2nd `  v
) )
9 fveq2 5848 . . . . . . 7  |-  ( t  =  w  ->  ( 2nd `  t )  =  ( 2nd `  w
) )
10 fvex 5858 . . . . . . 7  |-  ( 2nd `  w )  e.  _V
119, 3, 10fvmpt 5931 . . . . . 6  |-  ( w  e.  T  ->  ( F `  w )  =  ( 2nd `  w
) )
128, 11eqeqan12d 2477 . . . . 5  |-  ( ( v  e.  T  /\  w  e.  T )  ->  ( ( F `  v )  =  ( F `  w )  <-> 
( 2nd `  v
)  =  ( 2nd `  w ) ) )
1312adantl 464 . . . 4  |-  ( ( ( V USGrph  E  /\  N  e.  NN0 )  /\  ( v  e.  T  /\  w  e.  T
) )  ->  (
( F `  v
)  =  ( F `
 w )  <->  ( 2nd `  v )  =  ( 2nd `  w ) ) )
14 fveq2 5848 . . . . . . . . 9  |-  ( p  =  v  ->  ( 1st `  p )  =  ( 1st `  v
) )
1514fveq2d 5852 . . . . . . . 8  |-  ( p  =  v  ->  ( # `
 ( 1st `  p
) )  =  (
# `  ( 1st `  v ) ) )
1615eqeq1d 2456 . . . . . . 7  |-  ( p  =  v  ->  (
( # `  ( 1st `  p ) )  =  N  <->  ( # `  ( 1st `  v ) )  =  N ) )
1716, 1elrab2 3256 . . . . . 6  |-  ( v  e.  T  <->  ( v  e.  ( V Walks  E )  /\  ( # `  ( 1st `  v ) )  =  N ) )
18 fveq2 5848 . . . . . . . . 9  |-  ( p  =  w  ->  ( 1st `  p )  =  ( 1st `  w
) )
1918fveq2d 5852 . . . . . . . 8  |-  ( p  =  w  ->  ( # `
 ( 1st `  p
) )  =  (
# `  ( 1st `  w ) ) )
2019eqeq1d 2456 . . . . . . 7  |-  ( p  =  w  ->  (
( # `  ( 1st `  p ) )  =  N  <->  ( # `  ( 1st `  w ) )  =  N ) )
2120, 1elrab2 3256 . . . . . 6  |-  ( w  e.  T  <->  ( w  e.  ( V Walks  E )  /\  ( # `  ( 1st `  w ) )  =  N ) )
2217, 21anbi12i 695 . . . . 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 24914 . . . . . 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 1195 . . . . 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 473 . . . 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 2875 . 2  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  A. v  e.  T  A. w  e.  T  ( ( F `  v )  =  ( F `  w )  ->  v  =  w ) )
28 dff13 6141 . 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 662 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   class class class wbr 4439    |-> cmpt 4497   -->wf 5566   -1-1->wf1 5567   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   NN0cn0 10791   #chash 12390   USGrph cusg 24535   Walks cwalk 24703   WWalksN cwwlkn 24883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529  df-usgra 24538  df-wlk 24713  df-wwlk 24884  df-wwlkn 24885
This theorem is referenced by:  wlknwwlknbij  24918
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