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Theorem vdgrfval 21667
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
vdgrfval  |-  ( ( 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 } }
) ) ) )
Distinct variable groups:    x, u, A    u, E, x    u, V, x    u, W, x   
u, X, x

Proof of Theorem vdgrfval
Dummy variables  e 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-vdgr 21666 . . 3  |- VDeg  =  ( v  e.  _V , 
e  e.  _V  |->  ( u  e.  v  |->  ( ( # `  {
x  e.  dom  e  |  u  e.  (
e `  x ) } ) + e
( # `  { x  e.  dom  e  |  ( e `  x )  =  { u } } ) ) ) )
21a1i 11 . 2  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X )  -> VDeg  =  ( v  e. 
_V ,  e  e. 
_V  |->  ( u  e.  v  |->  ( ( # `  { x  e.  dom  e  |  u  e.  ( e `  x
) } ) + e ( # `  {
x  e.  dom  e  |  ( e `  x )  =  {
u } } ) ) ) ) )
3 simprl 734 . . 3  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
v  =  V )
4 simprr 735 . . . . . . . 8  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
e  =  E )
54dmeqd 5073 . . . . . . 7  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  ->  dom  e  =  dom  E )
6 simpl2 962 . . . . . . . 8  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  ->  E  Fn  A )
7 fndm 5545 . . . . . . . 8  |-  ( E  Fn  A  ->  dom  E  =  A )
86, 7syl 16 . . . . . . 7  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  ->  dom  E  =  A )
95, 8eqtrd 2469 . . . . . 6  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  ->  dom  e  =  A
)
104fveq1d 5731 . . . . . . 7  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
( e `  x
)  =  ( E `
 x ) )
1110eleq2d 2504 . . . . . 6  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
( u  e.  ( e `  x )  <-> 
u  e.  ( E `
 x ) ) )
129, 11rabeqbidv 2952 . . . . 5  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  ->  { x  e.  dom  e  |  u  e.  ( e `  x
) }  =  {
x  e.  A  |  u  e.  ( E `  x ) } )
1312fveq2d 5733 . . . 4  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
( # `  { x  e.  dom  e  |  u  e.  ( e `  x ) } )  =  ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } ) )
1410eqeq1d 2445 . . . . . 6  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
( ( e `  x )  =  {
u }  <->  ( E `  x )  =  {
u } ) )
159, 14rabeqbidv 2952 . . . . 5  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  ->  { x  e.  dom  e  |  ( e `  x )  =  {
u } }  =  { x  e.  A  |  ( E `  x )  =  {
u } } )
1615fveq2d 5733 . . . 4  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
( # `  { x  e.  dom  e  |  ( e `  x )  =  { u } } )  =  (
# `  { x  e.  A  |  ( E `  x )  =  { u } }
) )
1713, 16oveq12d 6100 . . 3  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
( ( # `  {
x  e.  dom  e  |  u  e.  (
e `  x ) } ) + e
( # `  { x  e.  dom  e  |  ( e `  x )  =  { u } } ) )  =  ( ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } ) + e ( # `  { x  e.  A  |  ( E `  x )  =  {
u } } ) ) )
183, 17mpteq12dv 4288 . 2  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
( u  e.  v 
|->  ( ( # `  {
x  e.  dom  e  |  u  e.  (
e `  x ) } ) + e
( # `  { x  e.  dom  e  |  ( e `  x )  =  { u } } ) ) )  =  ( u  e.  V  |->  ( ( # `  { x  e.  A  |  u  e.  ( E `  x ) } ) + e
( # `  { x  e.  A  |  ( E `  x )  =  { u } }
) ) ) )
19 elex 2965 . . 3  |-  ( V  e.  W  ->  V  e.  _V )
20193ad2ant1 979 . 2  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X )  ->  V  e.  _V )
21 fnex 5962 . . 3  |-  ( ( E  Fn  A  /\  A  e.  X )  ->  E  e.  _V )
22213adant1 976 . 2  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X )  ->  E  e.  _V )
23 mptexg 5966 . . 3  |-  ( V  e.  W  ->  (
u  e.  V  |->  ( ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } ) + e ( # `  { x  e.  A  |  ( E `  x )  =  {
u } } ) ) )  e.  _V )
24233ad2ant1 979 . 2  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X )  ->  ( u  e.  V  |->  ( ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } ) + e ( # `  { x  e.  A  |  ( E `  x )  =  {
u } } ) ) )  e.  _V )
252, 18, 20, 22, 24ovmpt2d 6202 1  |-  ( ( 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 } }
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {crab 2710   _Vcvv 2957   {csn 3815    e. cmpt 4267   dom cdm 4879    Fn wfn 5450   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084   + ecxad 10709   #chash 11619   VDeg cvdg 21665
This theorem is referenced by:  vdgrval  21668  vdgrf  21670  vdgrfif  21671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-vdgr 21666
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