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Theorem cusgrasizeindslem3 23382
Description: Lemma 3 for cusgrasizeinds 23383. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
Hypothesis
Ref Expression
cusgrares.f  |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x
) } )
Assertion
Ref Expression
cusgrasizeindslem3  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 { x  e. 
dom  E  |  N  e.  ( E `  x
) } )  =  ( ( # `  V
)  -  1 ) )
Distinct variable groups:    x, E    x, N    x, V
Allowed substitution hint:    F( x)

Proof of Theorem cusgrasizeindslem3
Dummy variables  n  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nbcusgra 23370 . . . 4  |-  ( ( V ComplUSGrph  E  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  =  ( V 
\  { N }
) )
213adant2 1007 . . 3  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  =  ( V 
\  { N }
) )
32fveq2d 5694 . 2  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  =  ( # `  ( V  \  { N }
) ) )
4 cusisusgra 23365 . . . . . 6  |-  ( V ComplUSGrph  E  ->  V USGrph  E )
54anim1i 568 . . . . 5  |-  ( ( V ComplUSGrph  E  /\  N  e.  V )  ->  ( V USGrph  E  /\  N  e.  V ) )
653adant2 1007 . . . 4  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( V USGrph  E  /\  N  e.  V ) )
7 nbgraf1o 23355 . . . 4  |-  ( ( V USGrph  E  /\  N  e.  V )  ->  E. f 
f : ( <. V ,  E >. Neighbors  N
)
-1-1-onto-> { x  e.  dom  E  |  N  e.  ( E `  x ) } )
86, 7syl 16 . . 3  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  E. f 
f : ( <. V ,  E >. Neighbors  N
)
-1-1-onto-> { x  e.  dom  E  |  N  e.  ( E `  x ) } )
9 nbusgra 23338 . . . . . . . 8  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  N )  =  {
n  e.  V  |  { N ,  n }  e.  ran  E } )
104, 9syl 16 . . . . . . 7  |-  ( V ComplUSGrph  E  ->  ( <. V ,  E >. Neighbors  N )  =  {
n  e.  V  |  { N ,  n }  e.  ran  E } )
1110adantr 465 . . . . . 6  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin )  ->  ( <. V ,  E >. Neighbors  N
)  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
12 rabfi 7536 . . . . . . 7  |-  ( V  e.  Fin  ->  { n  e.  V  |  { N ,  n }  e.  ran  E }  e.  Fin )
1312adantl 466 . . . . . 6  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin )  ->  { n  e.  V  |  { N ,  n }  e.  ran  E }  e.  Fin )
1411, 13eqeltrd 2516 . . . . 5  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin )  ->  ( <. V ,  E >. Neighbors  N
)  e.  Fin )
15143adant3 1008 . . . 4  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  e.  Fin )
16 usgrafis 23327 . . . . . . 7  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  E  e.  Fin )
174, 16sylan 471 . . . . . 6  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin )  ->  E  e.  Fin )
18 dmfi 7593 . . . . . . 7  |-  ( E  e.  Fin  ->  dom  E  e.  Fin )
19 rabfi 7536 . . . . . . 7  |-  ( dom 
E  e.  Fin  ->  { x  e.  dom  E  |  N  e.  ( E `  x ) }  e.  Fin )
2018, 19syl 16 . . . . . 6  |-  ( E  e.  Fin  ->  { x  e.  dom  E  |  N  e.  ( E `  x
) }  e.  Fin )
2117, 20syl 16 . . . . 5  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin )  ->  { x  e.  dom  E  |  N  e.  ( E `  x
) }  e.  Fin )
22213adant3 1008 . . . 4  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  { x  e.  dom  E  |  N  e.  ( E `  x
) }  e.  Fin )
23 hasheqf1o 12119 . . . 4  |-  ( ( ( <. V ,  E >. Neighbors  N )  e.  Fin  /\ 
{ x  e.  dom  E  |  N  e.  ( E `  x ) }  e.  Fin )  ->  ( ( # `  ( <. V ,  E >. Neighbors  N
) )  =  (
# `  { x  e.  dom  E  |  N  e.  ( E `  x
) } )  <->  E. f 
f : ( <. V ,  E >. Neighbors  N
)
-1-1-onto-> { x  e.  dom  E  |  N  e.  ( E `  x ) } ) )
2415, 22, 23syl2anc 661 . . 3  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  =  (
# `  { x  e.  dom  E  |  N  e.  ( E `  x
) } )  <->  E. f 
f : ( <. V ,  E >. Neighbors  N
)
-1-1-onto-> { x  e.  dom  E  |  N  e.  ( E `  x ) } ) )
258, 24mpbird 232 . 2  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  =  ( # `  {
x  e.  dom  E  |  N  e.  ( E `  x ) } ) )
26 hashdifsn 12168 . . 3  |-  ( ( V  e.  Fin  /\  N  e.  V )  ->  ( # `  ( V  \  { N }
) )  =  ( ( # `  V
)  -  1 ) )
27263adant1 1006 . 2  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( V  \  { N } ) )  =  ( ( # `  V )  -  1 ) )
283, 25, 273eqtr3d 2482 1  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 { x  e. 
dom  E  |  N  e.  ( E `  x
) } )  =  ( ( # `  V
)  -  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756    e/ wnel 2606   {crab 2718    \ cdif 3324   {csn 3876   {cpr 3878   <.cop 3882   class class class wbr 4291   dom cdm 4839   ran crn 4840    |` cres 4841   -1-1-onto->wf1o 5416   ` cfv 5417  (class class class)co 6090   Fincfn 7309   1c1 9282    - cmin 9594   #chash 12102   USGrph cusg 23263   Neighbors cnbgra 23328   ComplUSGrph ccusgra 23329
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-2o 6920  df-oadd 6923  df-er 7100  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-hash 12103  df-usgra 23265  df-nbgra 23331  df-cusgra 23332
This theorem is referenced by:  cusgrasizeinds  23383
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