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Theorem vdgrval 24719
Description: The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
Assertion
Ref Expression
vdgrval  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  U  e.  V )  ->  (
( V VDeg  E ) `  U )  =  ( ( # `  {
x  e.  A  |  U  e.  ( E `  x ) } ) +e ( # `  { x  e.  A  |  ( E `  x )  =  { U } } ) ) )
Distinct variable groups:    x, A    x, E    x, V    x, W    x, U    x, X

Proof of Theorem vdgrval
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 vdgrfval 24718 . . 3  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X )  ->  ( V VDeg  E )  =  ( u  e.  V  |->  ( ( # `  { x  e.  A  |  u  e.  ( E `  x ) } ) +e
( # `  { x  e.  A  |  ( E `  x )  =  { u } }
) ) ) )
21fveq1d 5874 . 2  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X )  ->  ( ( V VDeg  E
) `  U )  =  ( ( u  e.  V  |->  ( (
# `  { x  e.  A  |  u  e.  ( E `  x
) } ) +e ( # `  {
x  e.  A  | 
( E `  x
)  =  { u } } ) ) ) `
 U ) )
3 eleq1 2539 . . . . . 6  |-  ( u  =  U  ->  (
u  e.  ( E `
 x )  <->  U  e.  ( E `  x ) ) )
43rabbidv 3110 . . . . 5  |-  ( u  =  U  ->  { x  e.  A  |  u  e.  ( E `  x
) }  =  {
x  e.  A  |  U  e.  ( E `  x ) } )
54fveq2d 5876 . . . 4  |-  ( u  =  U  ->  ( # `
 { x  e.  A  |  u  e.  ( E `  x
) } )  =  ( # `  {
x  e.  A  |  U  e.  ( E `  x ) } ) )
6 sneq 4043 . . . . . . 7  |-  ( u  =  U  ->  { u }  =  { U } )
76eqeq2d 2481 . . . . . 6  |-  ( u  =  U  ->  (
( E `  x
)  =  { u } 
<->  ( E `  x
)  =  { U } ) )
87rabbidv 3110 . . . . 5  |-  ( u  =  U  ->  { x  e.  A  |  ( E `  x )  =  { u } }  =  { x  e.  A  |  ( E `  x )  =  { U } } )
98fveq2d 5876 . . . 4  |-  ( u  =  U  ->  ( # `
 { x  e.  A  |  ( E `
 x )  =  { u } }
)  =  ( # `  { x  e.  A  |  ( E `  x )  =  { U } } ) )
105, 9oveq12d 6313 . . 3  |-  ( u  =  U  ->  (
( # `  { x  e.  A  |  u  e.  ( E `  x
) } ) +e ( # `  {
x  e.  A  | 
( E `  x
)  =  { u } } ) )  =  ( ( # `  {
x  e.  A  |  U  e.  ( E `  x ) } ) +e ( # `  { x  e.  A  |  ( E `  x )  =  { U } } ) ) )
11 eqid 2467 . . 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 } }
) ) )
12 ovex 6320 . . 3  |-  ( (
# `  { x  e.  A  |  U  e.  ( E `  x
) } ) +e ( # `  {
x  e.  A  | 
( E `  x
)  =  { U } } ) )  e. 
_V
1310, 11, 12fvmpt 5957 . 2  |-  ( U  e.  V  ->  (
( u  e.  V  |->  ( ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } ) +e ( # `  { x  e.  A  |  ( E `  x )  =  {
u } } ) ) ) `  U
)  =  ( (
# `  { x  e.  A  |  U  e.  ( E `  x
) } ) +e ( # `  {
x  e.  A  | 
( E `  x
)  =  { U } } ) ) )
142, 13sylan9eq 2528 1  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  U  e.  V )  ->  (
( V VDeg  E ) `  U )  =  ( ( # `  {
x  e.  A  |  U  e.  ( E `  x ) } ) +e ( # `  { x  e.  A  |  ( E `  x )  =  { U } } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2821   {csn 4033    |-> cmpt 4511    Fn wfn 5589   ` cfv 5594  (class class class)co 6295   +ecxad 11328   #chash 12385   VDeg cvdg 24716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-vdgr 24717
This theorem is referenced by:  vdgrfival  24720  vdgr0  24723  vdgrun  24724  vdgr1d  24726  vdgr1b  24727  vdgr1a  24729  vdusgraval  24730
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