MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cusgrasizeindslem3 Structured version   Unicode version

Theorem cusgrasizeindslem3 24139
Description: Lemma 3 for cusgrasizeinds 24140. (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 24127 . . . 4  |-  ( ( V ComplUSGrph  E  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  =  ( V 
\  { N }
) )
213adant2 1010 . . 3  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  =  ( V 
\  { N }
) )
32fveq2d 5863 . 2  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  =  ( # `  ( V  \  { N }
) ) )
4 cusisusgra 24122 . . . . . 6  |-  ( V ComplUSGrph  E  ->  V USGrph  E )
54anim1i 568 . . . . 5  |-  ( ( V ComplUSGrph  E  /\  N  e.  V )  ->  ( V USGrph  E  /\  N  e.  V ) )
653adant2 1010 . . . 4  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( V USGrph  E  /\  N  e.  V ) )
7 nbgraf1o 24111 . . . 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 24092 . . . . . . . 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 7736 . . . . . . 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 2550 . . . . 5  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin )  ->  ( <. V ,  E >. Neighbors  N
)  e.  Fin )
15143adant3 1011 . . . 4  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  e.  Fin )
16 usgrafis 24079 . . . . . . 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 7794 . . . . . . 7  |-  ( E  e.  Fin  ->  dom  E  e.  Fin )
19 rabfi 7736 . . . . . . 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 1011 . . . 4  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  { x  e.  dom  E  |  N  e.  ( E `  x
) }  e.  Fin )
23 hasheqf1o 12379 . . . 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 12431 . . 3  |-  ( ( V  e.  Fin  /\  N  e.  V )  ->  ( # `  ( V  \  { N }
) )  =  ( ( # `  V
)  -  1 ) )
27263adant1 1009 . 2  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( V  \  { N } ) )  =  ( ( # `  V )  -  1 ) )
283, 25, 273eqtr3d 2511 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 968    = wceq 1374   E.wex 1591    e. wcel 1762    e/ wnel 2658   {crab 2813    \ cdif 3468   {csn 4022   {cpr 4024   <.cop 4028   class class class wbr 4442   dom cdm 4994   ran crn 4995    |` cres 4996   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6277   Fincfn 7508   1c1 9484    - cmin 9796   #chash 12362   USGrph cusg 23995   Neighbors cnbgra 24081   ComplUSGrph ccusgra 24082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-hash 12363  df-usgra 23998  df-nbgra 24084  df-cusgra 24085
This theorem is referenced by:  cusgrasizeinds  24140
  Copyright terms: Public domain W3C validator