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

Theorem vdgrfif 21623
Description: The vertex degree function on graphs of finite size is a function from vertices to nonnegative integers. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
Assertion
Ref Expression
vdgrfif  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  ->  ( V VDeg  E ) : V --> NN0 )

Proof of Theorem vdgrfif
Dummy variables  u  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabfi 7292 . . . . . . . . 9  |-  ( A  e.  Fin  ->  { x  e.  A  |  u  e.  ( E `  x
) }  e.  Fin )
2 hashcl 11594 . . . . . . . . 9  |-  ( { x  e.  A  |  u  e.  ( E `  x ) }  e.  Fin  ->  ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } )  e.  NN0 )
3 nn0re 10186 . . . . . . . . 9  |-  ( (
# `  { x  e.  A  |  u  e.  ( E `  x
) } )  e. 
NN0  ->  ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } )  e.  RR )
41, 2, 33syl 19 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( # `
 { x  e.  A  |  u  e.  ( E `  x
) } )  e.  RR )
5 rabfi 7292 . . . . . . . . 9  |-  ( A  e.  Fin  ->  { x  e.  A  |  ( E `  x )  =  { u } }  e.  Fin )
6 hashcl 11594 . . . . . . . . 9  |-  ( { x  e.  A  | 
( E `  x
)  =  { u } }  e.  Fin  ->  ( # `  {
x  e.  A  | 
( E `  x
)  =  { u } } )  e.  NN0 )
7 nn0re 10186 . . . . . . . . 9  |-  ( (
# `  { x  e.  A  |  ( E `  x )  =  { u } }
)  e.  NN0  ->  (
# `  { x  e.  A  |  ( E `  x )  =  { u } }
)  e.  RR )
85, 6, 73syl 19 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( # `
 { x  e.  A  |  ( E `
 x )  =  { u } }
)  e.  RR )
94, 8jca 519 . . . . . . 7  |-  ( A  e.  Fin  ->  (
( # `  { x  e.  A  |  u  e.  ( E `  x
) } )  e.  RR  /\  ( # `  { x  e.  A  |  ( E `  x )  =  {
u } } )  e.  RR ) )
1093ad2ant3 980 . . . . . 6  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  ->  ( ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } )  e.  RR  /\  ( # `
 { x  e.  A  |  ( E `
 x )  =  { u } }
)  e.  RR ) )
1110adantr 452 . . . . 5  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  /\  u  e.  V
)  ->  ( ( # `
 { x  e.  A  |  u  e.  ( E `  x
) } )  e.  RR  /\  ( # `  { x  e.  A  |  ( E `  x )  =  {
u } } )  e.  RR ) )
12 rexadd 10774 . . . . 5  |-  ( ( ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } )  e.  RR  /\  ( # `
 { x  e.  A  |  ( E `
 x )  =  { u } }
)  e.  RR )  ->  ( ( # `  { x  e.  A  |  u  e.  ( E `  x ) } ) + e
( # `  { x  e.  A  |  ( E `  x )  =  { u } }
) )  =  ( ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } )  +  ( # `  {
x  e.  A  | 
( E `  x
)  =  { u } } ) ) )
1311, 12syl 16 . . . 4  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  /\  u  e.  V
)  ->  ( ( # `
 { x  e.  A  |  u  e.  ( E `  x
) } ) + e ( # `  {
x  e.  A  | 
( E `  x
)  =  { u } } ) )  =  ( ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } )  +  ( # `  {
x  e.  A  | 
( E `  x
)  =  { u } } ) ) )
14 simpl3 962 . . . . . 6  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  /\  u  e.  V
)  ->  A  e.  Fin )
1514, 1, 23syl 19 . . . . 5  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  /\  u  e.  V
)  ->  ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } )  e.  NN0 )
1614, 5, 63syl 19 . . . . 5  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  /\  u  e.  V
)  ->  ( # `  {
x  e.  A  | 
( E `  x
)  =  { u } } )  e.  NN0 )
1715, 16nn0addcld 10234 . . . 4  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  /\  u  e.  V
)  ->  ( ( # `
 { x  e.  A  |  u  e.  ( E `  x
) } )  +  ( # `  {
x  e.  A  | 
( E `  x
)  =  { u } } ) )  e. 
NN0 )
1813, 17eqeltrd 2478 . . 3  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  /\  u  e.  V
)  ->  ( ( # `
 { x  e.  A  |  u  e.  ( E `  x
) } ) + e ( # `  {
x  e.  A  | 
( E `  x
)  =  { u } } ) )  e. 
NN0 )
19 eqid 2404 . . 3  |-  ( u  e.  V  |->  ( (
# `  { x  e.  A  |  u  e.  ( E `  x
) } ) + e ( # `  {
x  e.  A  | 
( E `  x
)  =  { u } } ) ) )  =  ( u  e.  V  |->  ( ( # `  { x  e.  A  |  u  e.  ( E `  x ) } ) + e
( # `  { x  e.  A  |  ( E `  x )  =  { u } }
) ) )
2018, 19fmptd 5852 . 2  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  ->  ( u  e.  V  |->  ( ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } ) + e ( # `  { x  e.  A  |  ( E `  x )  =  {
u } } ) ) ) : V --> NN0 )
21 vdgrfval 21619 . . 3  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  ->  ( V VDeg  E )  =  ( u  e.  V  |->  ( ( # `  { x  e.  A  |  u  e.  ( E `  x ) } ) + e
( # `  { x  e.  A  |  ( E `  x )  =  { u } }
) ) ) )
2221feq1d 5539 . 2  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  ->  ( ( V VDeg  E
) : V --> NN0  <->  ( u  e.  V  |->  ( (
# `  { x  e.  A  |  u  e.  ( E `  x
) } ) + e ( # `  {
x  e.  A  | 
( E `  x
)  =  { u } } ) ) ) : V --> NN0 )
)
2320, 22mpbird 224 1  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  ->  ( V VDeg  E ) : V --> NN0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {crab 2670   {csn 3774    e. cmpt 4226    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   RRcr 8945    + caddc 8949   NN0cn0 10177   + ecxad 10664   #chash 11573   VDeg cvdg 21617
This theorem is referenced by:  eupath2lem3  21654  vdegp1ai  21659  usgfidegfi  28090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-xadd 10667  df-hash 11574  df-vdgr 21618
  Copyright terms: Public domain W3C validator