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Theorem vdgrfval 21627
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 21626 . . 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 733 . . 3  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
v  =  V )
4 simprr 734 . . . . . . . 8  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
e  =  E )
54dmeqd 5039 . . . . . . 7  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  ->  dom  e  =  dom  E )
6 simpl2 961 . . . . . . . 8  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  ->  E  Fn  A )
7 fndm 5511 . . . . . . . 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 2444 . . . . . 6  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  ->  dom  e  =  A
)
104fveq1d 5697 . . . . . . 7  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
( e `  x
)  =  ( E `
 x ) )
1110eleq2d 2479 . . . . . 6  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
( u  e.  ( e `  x )  <-> 
u  e.  ( E `
 x ) ) )
129, 11rabeqbidv 2919 . . . . 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 5699 . . . 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 2420 . . . . . 6  |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
( ( e `  x )  =  {
u }  <->  ( E `  x )  =  {
u } ) )
159, 14rabeqbidv 2919 . . . . 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 5699 . . . 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 6066 . . 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 4255 . 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 2932 . . 3  |-  ( V  e.  W  ->  V  e.  _V )
20193ad2ant1 978 . 2  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X )  ->  V  e.  _V )
21 fnex 5928 . . 3  |-  ( ( E  Fn  A  /\  A  e.  X )  ->  E  e.  _V )
22213adant1 975 . 2  |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X )  ->  E  e.  _V )
23 mptexg 5932 . . 3  |-  ( V  e.  W  ->  (
u  e.  V  |->  ( ( # `  {
x  e.  A  |  u  e.  ( E `  x ) } ) + e ( # `  { x  e.  A  |  ( E `  x )  =  {
u } } ) ) )  e.  _V )
24233ad2ant1 978 . 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 6168 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {crab 2678   _Vcvv 2924   {csn 3782    e. cmpt 4234   dom cdm 4845    Fn wfn 5416   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   + ecxad 10672   #chash 11581   VDeg cvdg 21625
This theorem is referenced by:  vdgrval  21628  vdgrf  21630  vdgrfif  21631
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-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-vdgr 21626
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