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Theorem vdgrfif 23520
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 7529 . . . . . . . . 9  |-  ( A  e.  Fin  ->  { x  e.  A  |  u  e.  ( E `  x
) }  e.  Fin )
2 hashcl 12118 . . . . . . . . 9  |-  ( { x  e.  A  |  u  e.  ( E `  x ) }  e.  Fin  ->  ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } )  e.  NN0 )
3 nn0re 10580 . . . . . . . . 9  |-  ( (
# `  { x  e.  A  |  u  e.  ( E `  x
) } )  e. 
NN0  ->  ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } )  e.  RR )
41, 2, 33syl 20 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( # `
 { x  e.  A  |  u  e.  ( E `  x
) } )  e.  RR )
5 rabfi 7529 . . . . . . . . 9  |-  ( A  e.  Fin  ->  { x  e.  A  |  ( E `  x )  =  { u } }  e.  Fin )
6 hashcl 12118 . . . . . . . . 9  |-  ( { x  e.  A  | 
( E `  x
)  =  { u } }  e.  Fin  ->  ( # `  {
x  e.  A  | 
( E `  x
)  =  { u } } )  e.  NN0 )
7 nn0re 10580 . . . . . . . . 9  |-  ( (
# `  { x  e.  A  |  ( E `  x )  =  { u } }
)  e.  NN0  ->  (
# `  { x  e.  A  |  ( E `  x )  =  { u } }
)  e.  RR )
85, 6, 73syl 20 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( # `
 { x  e.  A  |  ( E `
 x )  =  { u } }
)  e.  RR )
94, 8jca 532 . . . . . . 7  |-  ( A  e.  Fin  ->  (
( # `  { x  e.  A  |  u  e.  ( E `  x
) } )  e.  RR  /\  ( # `  { x  e.  A  |  ( E `  x )  =  {
u } } )  e.  RR ) )
1093ad2ant3 1011 . . . . . 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 465 . . . . 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 11194 . . . . 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 993 . . . . . 6  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  /\  u  e.  V
)  ->  A  e.  Fin )
1514, 1, 23syl 20 . . . . 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 20 . . . . 5  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  /\  u  e.  V
)  ->  ( # `  {
x  e.  A  | 
( E `  x
)  =  { u } } )  e.  NN0 )
1715, 16nn0addcld 10632 . . . 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 2512 . . 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 2438 . . 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 5862 . 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 23516 . . 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 5541 . 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 232 1  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  ->  ( V VDeg  E ) : V --> NN0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {crab 2714   {csn 3872    e. cmpt 4345    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   Fincfn 7302   RRcr 9273    + caddc 9277   NN0cn0 10571   +ecxad 11079   #chash 12095   VDeg cvdg 23514
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-xadd 11082  df-hash 12096  df-vdgr 23515
This theorem is referenced by:  eupath2lem3  23551  vdegp1ai  23556  usgfidegfi  30480
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