Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wlkiswwlk2lem5 Structured version   Unicode version

Theorem wlkiswwlk2lem5 30476
Description: Lemma 5 for wlkiswwlk2 30478. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
Hypothesis
Ref Expression
wlkiswwlk2lem.f  |-  F  =  ( x  e.  ( 0..^ ( ( # `  P )  -  1 ) )  |->  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1
) ) } ) )
Assertion
Ref Expression
wlkiswwlk2lem5  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  ( A. i  e.  ( 0..^ ( (
# `  P )  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  ->  F  e. Word  dom  E
) )
Distinct variable groups:    i, E, x    i, F    P, i, x    i, V, x
Allowed substitution hint:    F( x)

Proof of Theorem wlkiswwlk2lem5
StepHypRef Expression
1 usgraf1o 23432 . . . . . . 7  |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran 
E )
213ad2ant1 1009 . . . . . 6  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  E : dom  E -1-1-onto-> ran 
E )
32ad2antrr 725 . . . . 5  |-  ( ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E )  /\  x  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )  ->  E : dom  E -1-1-onto-> ran  E )
4 simpr 461 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  x  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )  ->  x  e.  ( 0..^ ( ( # `  P )  -  1 ) ) )
5 fveq2 5798 . . . . . . . . . . 11  |-  ( i  =  x  ->  ( P `  i )  =  ( P `  x ) )
6 oveq1 6206 . . . . . . . . . . . 12  |-  ( i  =  x  ->  (
i  +  1 )  =  ( x  + 
1 ) )
76fveq2d 5802 . . . . . . . . . . 11  |-  ( i  =  x  ->  ( P `  ( i  +  1 ) )  =  ( P `  ( x  +  1
) ) )
85, 7preq12d 4069 . . . . . . . . . 10  |-  ( i  =  x  ->  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  =  { ( P `  x ) ,  ( P `  ( x  +  1 ) ) } )
98eleq1d 2523 . . . . . . . . 9  |-  ( i  =  x  ->  ( { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( P `  x
) ,  ( P `
 ( x  + 
1 ) ) }  e.  ran  E ) )
109adantl 466 . . . . . . . 8  |-  ( ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  x  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )  /\  i  =  x )  ->  ( { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( P `  x
) ,  ( P `
 ( x  + 
1 ) ) }  e.  ran  E ) )
114, 10rspcdv 3180 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  x  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )  ->  ( A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E  ->  { ( P `
 x ) ,  ( P `  (
x  +  1 ) ) }  e.  ran  E ) )
1211impancom 440 . . . . . 6  |-  ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E )  ->  ( x  e.  ( 0..^ ( (
# `  P )  -  1 ) )  ->  { ( P `
 x ) ,  ( P `  (
x  +  1 ) ) }  e.  ran  E ) )
1312imp 429 . . . . 5  |-  ( ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E )  /\  x  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )  ->  { ( P `  x ) ,  ( P `  ( x  +  1
) ) }  e.  ran  E )
14 f1ocnvdm 6097 . . . . 5  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { ( P `
 x ) ,  ( P `  (
x  +  1 ) ) }  e.  ran  E )  ->  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1
) ) } )  e.  dom  E )
153, 13, 14syl2anc 661 . . . 4  |-  ( ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E )  /\  x  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )  ->  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1
) ) } )  e.  dom  E )
16 wlkiswwlk2lem.f . . . 4  |-  F  =  ( x  e.  ( 0..^ ( ( # `  P )  -  1 ) )  |->  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1
) ) } ) )
1715, 16fmptd 5975 . . 3  |-  ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E )  ->  F :
( 0..^ ( (
# `  P )  -  1 ) ) --> dom  E )
18 iswrdi 12356 . . 3  |-  ( F : ( 0..^ ( ( # `  P
)  -  1 ) ) --> dom  E  ->  F  e. Word  dom  E )
1917, 18syl 16 . 2  |-  ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E )  ->  F  e. Word  dom 
E )
2019ex 434 1  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  ( A. i  e.  ( 0..^ ( (
# `  P )  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  ->  F  e. Word  dom  E
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2798   {cpr 3986   class class class wbr 4399    |-> cmpt 4457   `'ccnv 4946   dom cdm 4947   ran crn 4948   -->wf 5521   -1-1-onto->wf1o 5524   ` cfv 5525  (class class class)co 6199   0cc0 9392   1c1 9393    + caddc 9395    <_ cle 9529    - cmin 9705  ..^cfzo 11664   #chash 12219  Word cword 12338   USGrph cusg 23415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-fzo 11665  df-hash 12220  df-word 12346  df-usgra 23417
This theorem is referenced by:  wlkiswwlk2lem6  30477
  Copyright terms: Public domain W3C validator