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Theorem rusgranumwlkl1lem1 30701
Description: Lemma for rusgranumwlkl1 30702. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
Assertion
Ref Expression
rusgranumwlkl1lem1  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( # `  {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) } )  =  K )
Distinct variable groups:    w, E    w, P    w, V
Allowed substitution hint:    K( w)

Proof of Theorem rusgranumwlkl1lem1
Dummy variables  f  p  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rusisusgra 30691 . . . . . 6  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
2 usgrav 23417 . . . . . . 7  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
32simpld 459 . . . . . 6  |-  ( V USGrph  E  ->  V  e.  _V )
41, 3syl 16 . . . . 5  |-  ( <. V ,  E >. RegUSGrph  K  ->  V  e.  _V )
54adantr 465 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  V  e.  _V )
6 wrdexg 12357 . . . . . 6  |-  ( V  e.  _V  -> Word  V  e. 
_V )
7 rabexg 4545 . . . . . 6  |-  (Word  V  e.  _V  ->  { w  e. Word  V  |  ( (
# `  w )  =  2  /\  (
w `  0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  ran  E ) }  e.  _V )
86, 7syl 16 . . . . 5  |-  ( V  e.  _V  ->  { w  e. Word  V  |  ( (
# `  w )  =  2  /\  (
w `  0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  ran  E ) }  e.  _V )
9 rabexg 4545 . . . . 5  |-  ( V  e.  _V  ->  { s  e.  V  |  { P ,  s }  e.  ran  E }  e.  _V )
108, 9jca 532 . . . 4  |-  ( V  e.  _V  ->  ( { w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) }  e.  _V  /\  { s  e.  V  |  { P ,  s }  e.  ran  E }  e.  _V ) )
115, 10syl 16 . . 3  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) }  e.  _V  /\  { s  e.  V  |  { P ,  s }  e.  ran  E }  e.  _V ) )
12 wrd2f1tovbij 30398 . . . 4  |-  ( ( V  e.  _V  /\  P  e.  V )  ->  E. f  f : { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E ) } -1-1-onto-> { s  e.  V  |  { P ,  s }  e.  ran  E } )
134, 12sylan 471 . . 3  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  E. f 
f : { w  e. Word  V  |  ( (
# `  w )  =  2  /\  (
w `  0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  ran  E ) } -1-1-onto-> { s  e.  V  |  { P ,  s }  e.  ran  E } )
14 hasheqf1oi 12234 . . 3  |-  ( ( { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E ) }  e.  _V  /\ 
{ s  e.  V  |  { P ,  s }  e.  ran  E }  e.  _V )  ->  ( E. f  f : { w  e. Word  V  |  ( ( # `
 w )  =  2  /\  ( w `
 0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  ran  E ) } -1-1-onto-> { s  e.  V  |  { P ,  s }  e.  ran  E }  ->  ( # `  {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) } )  =  (
# `  { s  e.  V  |  { P ,  s }  e.  ran  E } ) ) )
1511, 13, 14sylc 60 . 2  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( # `  {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) } )  =  (
# `  { s  e.  V  |  { P ,  s }  e.  ran  E } ) )
16 rusgraprop3 30698 . . . 4  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. p  e.  V  ( # `
 { s  e.  V  |  { p ,  s }  e.  ran  E } )  =  K ) )
17 preq1 4057 . . . . . . . . . . 11  |-  ( p  =  P  ->  { p ,  s }  =  { P ,  s } )
1817eleq1d 2521 . . . . . . . . . 10  |-  ( p  =  P  ->  ( { p ,  s }  e.  ran  E  <->  { P ,  s }  e.  ran  E ) )
1918rabbidv 3064 . . . . . . . . 9  |-  ( p  =  P  ->  { s  e.  V  |  {
p ,  s }  e.  ran  E }  =  { s  e.  V  |  { P ,  s }  e.  ran  E } )
2019fveq2d 5798 . . . . . . . 8  |-  ( p  =  P  ->  ( # `
 { s  e.  V  |  { p ,  s }  e.  ran  E } )  =  ( # `  {
s  e.  V  |  { P ,  s }  e.  ran  E }
) )
2120eqeq1d 2454 . . . . . . 7  |-  ( p  =  P  ->  (
( # `  { s  e.  V  |  {
p ,  s }  e.  ran  E }
)  =  K  <->  ( # `  {
s  e.  V  |  { P ,  s }  e.  ran  E }
)  =  K ) )
2221rspcva 3171 . . . . . 6  |-  ( ( P  e.  V  /\  A. p  e.  V  (
# `  { s  e.  V  |  {
p ,  s }  e.  ran  E }
)  =  K )  ->  ( # `  {
s  e.  V  |  { P ,  s }  e.  ran  E }
)  =  K )
2322expcom 435 . . . . 5  |-  ( A. p  e.  V  ( # `
 { s  e.  V  |  { p ,  s }  e.  ran  E } )  =  K  ->  ( P  e.  V  ->  ( # `  { s  e.  V  |  { P ,  s }  e.  ran  E } )  =  K ) )
24233ad2ant3 1011 . . . 4  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. p  e.  V  ( # `  {
s  e.  V  |  { p ,  s }  e.  ran  E } )  =  K )  ->  ( P  e.  V  ->  ( # `  { s  e.  V  |  { P ,  s }  e.  ran  E } )  =  K ) )
2516, 24syl 16 . . 3  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( P  e.  V  ->  ( # `  {
s  e.  V  |  { P ,  s }  e.  ran  E }
)  =  K ) )
2625imp 429 . 2  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( # `  {
s  e.  V  |  { P ,  s }  e.  ran  E }
)  =  K )
2715, 26eqtrd 2493 1  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( # `  {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) } )  =  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   A.wral 2796   {crab 2800   _Vcvv 3072   {cpr 3982   <.cop 3986   class class class wbr 4395   ran crn 4944   -1-1-onto->wf1o 5520   ` cfv 5521   0cc0 9388   1c1 9389   2c2 10477   NN0cn0 10685   #chash 12215  Word cword 12334   USGrph cusg 23411   RegUSGrph crusgra 30683
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-map 7321  df-pm 7322  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-n0 10686  df-z 10753  df-uz 10968  df-xadd 11196  df-fz 11550  df-fzo 11661  df-hash 12216  df-word 12342  df-usgra 23413  df-nbgra 23479  df-vdgr 23711  df-rgra 30684  df-rusgra 30685
This theorem is referenced by:  rusgranumwlkl1  30702  numclwwlkovf2num  30821
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