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Theorem cusgrasizeindslem3 24879
Description: Lemma 3 for cusgrasizeinds 24880. (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 24867 . . . 4  |-  ( ( V ComplUSGrph  E  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  =  ( V 
\  { N }
) )
213adant2 1016 . . 3  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  =  ( V 
\  { N }
) )
32fveq2d 5852 . 2  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  =  ( # `  ( V  \  { N }
) ) )
4 cusisusgra 24862 . . . . . 6  |-  ( V ComplUSGrph  E  ->  V USGrph  E )
54anim1i 566 . . . . 5  |-  ( ( V ComplUSGrph  E  /\  N  e.  V )  ->  ( V USGrph  E  /\  N  e.  V ) )
653adant2 1016 . . . 4  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( V USGrph  E  /\  N  e.  V ) )
7 nbgraf1o 24851 . . . 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 17 . . 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 24832 . . . . . . . 8  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  N )  =  {
n  e.  V  |  { N ,  n }  e.  ran  E } )
104, 9syl 17 . . . . . . 7  |-  ( V ComplUSGrph  E  ->  ( <. V ,  E >. Neighbors  N )  =  {
n  e.  V  |  { N ,  n }  e.  ran  E } )
1110adantr 463 . . . . . 6  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin )  ->  ( <. V ,  E >. Neighbors  N
)  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
12 rabfi 7778 . . . . . . 7  |-  ( V  e.  Fin  ->  { n  e.  V  |  { N ,  n }  e.  ran  E }  e.  Fin )
1312adantl 464 . . . . . 6  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin )  ->  { n  e.  V  |  { N ,  n }  e.  ran  E }  e.  Fin )
1411, 13eqeltrd 2490 . . . . 5  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin )  ->  ( <. V ,  E >. Neighbors  N
)  e.  Fin )
15143adant3 1017 . . . 4  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  e.  Fin )
16 usgrafis 24819 . . . . . . 7  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  E  e.  Fin )
174, 16sylan 469 . . . . . 6  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin )  ->  E  e.  Fin )
18 dmfi 7836 . . . . . . 7  |-  ( E  e.  Fin  ->  dom  E  e.  Fin )
19 rabfi 7778 . . . . . . 7  |-  ( dom 
E  e.  Fin  ->  { x  e.  dom  E  |  N  e.  ( E `  x ) }  e.  Fin )
2018, 19syl 17 . . . . . 6  |-  ( E  e.  Fin  ->  { x  e.  dom  E  |  N  e.  ( E `  x
) }  e.  Fin )
2117, 20syl 17 . . . . 5  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin )  ->  { x  e.  dom  E  |  N  e.  ( E `  x
) }  e.  Fin )
22213adant3 1017 . . . 4  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  { x  e.  dom  E  |  N  e.  ( E `  x
) }  e.  Fin )
23 hasheqf1o 12467 . . . 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 659 . . 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 12524 . . 3  |-  ( ( V  e.  Fin  /\  N  e.  V )  ->  ( # `  ( V  \  { N }
) )  =  ( ( # `  V
)  -  1 ) )
27263adant1 1015 . 2  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( V  \  { N } ) )  =  ( ( # `  V )  -  1 ) )
283, 25, 273eqtr3d 2451 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 367    /\ w3a 974    = wceq 1405   E.wex 1633    e. wcel 1842    e/ wnel 2599   {crab 2757    \ cdif 3410   {csn 3971   {cpr 3973   <.cop 3977   class class class wbr 4394   dom cdm 4822   ran crn 4823    |` cres 4824   -1-1-onto->wf1o 5567   ` cfv 5568  (class class class)co 6277   Fincfn 7553   1c1 9522    - cmin 9840   #chash 12450   USGrph cusg 24734   Neighbors cnbgra 24821   ComplUSGrph ccusgra 24822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-hash 12451  df-usgra 24737  df-nbgra 24824  df-cusgra 24825
This theorem is referenced by:  cusgrasizeinds  24880
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