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Theorem vdusgraval 25034
Description: The value of the vertex degree function for simple undirected graphs. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
Assertion
Ref Expression
vdusgraval  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  (
( V VDeg  E ) `  U )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) )
Distinct variable groups:    x, E    x, U    x, V

Proof of Theorem vdusgraval
StepHypRef Expression
1 usgranloop0 24507 . 2  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/) )
2 usgrafun 24476 . . . . . . 7  |-  ( V USGrph  E  ->  Fun  E )
3 usgrav 24465 . . . . . . 7  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
4 funfn 5623 . . . . . . . 8  |-  ( Fun 
E  <->  E  Fn  dom  E )
5 simpl 457 . . . . . . . . . . 11  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  V  e.  _V )
65adantl 466 . . . . . . . . . 10  |-  ( ( E  Fn  dom  E  /\  ( V  e.  _V  /\  E  e.  _V )
)  ->  V  e.  _V )
7 simpl 457 . . . . . . . . . 10  |-  ( ( E  Fn  dom  E  /\  ( V  e.  _V  /\  E  e.  _V )
)  ->  E  Fn  dom  E )
8 dmexg 6730 . . . . . . . . . . . 12  |-  ( E  e.  _V  ->  dom  E  e.  _V )
98adantl 466 . . . . . . . . . . 11  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  dom  E  e.  _V )
109adantl 466 . . . . . . . . . 10  |-  ( ( E  Fn  dom  E  /\  ( V  e.  _V  /\  E  e.  _V )
)  ->  dom  E  e. 
_V )
116, 7, 103jca 1176 . . . . . . . . 9  |-  ( ( E  Fn  dom  E  /\  ( V  e.  _V  /\  E  e.  _V )
)  ->  ( V  e.  _V  /\  E  Fn  dom  E  /\  dom  E  e.  _V ) )
1211ex 434 . . . . . . . 8  |-  ( E  Fn  dom  E  -> 
( ( V  e. 
_V  /\  E  e.  _V )  ->  ( V  e.  _V  /\  E  Fn  dom  E  /\  dom  E  e.  _V ) ) )
134, 12sylbi 195 . . . . . . 7  |-  ( Fun 
E  ->  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V  e.  _V  /\  E  Fn  dom  E  /\  dom  E  e.  _V ) ) )
142, 3, 13sylc 60 . . . . . 6  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  Fn  dom  E  /\  dom  E  e.  _V ) )
1514anim1i 568 . . . . 5  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  (
( V  e.  _V  /\  E  Fn  dom  E  /\  dom  E  e.  _V )  /\  U  e.  V
) )
16 vdgrval 25023 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  Fn  dom  E  /\  dom  E  e.  _V )  /\  U  e.  V
)  ->  ( ( V VDeg  E ) `  U
)  =  ( (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) +e ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } ) ) )
17 fveq2 5872 . . . . . . . . . . 11  |-  ( { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/)  ->  ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } )  =  ( # `  (/) ) )
18 hash0 12440 . . . . . . . . . . 11  |-  ( # `  (/) )  =  0
1917, 18syl6eq 2514 . . . . . . . . . 10  |-  ( { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/)  ->  ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } )  =  0 )
20 oveq2 6304 . . . . . . . . . . . . 13  |-  ( (
# `  { x  e.  dom  E  |  ( E `  x )  =  { U } } )  =  0  ->  ( ( # `  { x  e.  dom  E  |  U  e.  ( E `  x ) } ) +e
( # `  { x  e.  dom  E  |  ( E `  x )  =  { U } } ) )  =  ( ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) +e 0 ) )
2120adantr 465 . . . . . . . . . . . 12  |-  ( ( ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } )  =  0  /\  V USGrph  E
)  ->  ( ( # `
 { x  e. 
dom  E  |  U  e.  ( E `  x
) } ) +e ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } ) )  =  ( ( # `  { x  e.  dom  E  |  U  e.  ( E `  x ) } ) +e 0 ) )
223simprd 463 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  E  e.  _V )
23 rabexg 4606 . . . . . . . . . . . . . . 15  |-  ( dom 
E  e.  _V  ->  { x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  _V )
2422, 8, 233syl 20 . . . . . . . . . . . . . 14  |-  ( V USGrph  E  ->  { x  e. 
dom  E  |  U  e.  ( E `  x
) }  e.  _V )
2524adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } )  =  0  /\  V USGrph  E
)  ->  { x  e.  dom  E  |  U  e.  ( E `  x
) }  e.  _V )
26 hashxrcl 12432 . . . . . . . . . . . . 13  |-  ( { x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  _V  ->  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } )  e. 
RR* )
27 xaddid1 11463 . . . . . . . . . . . . 13  |-  ( (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } )  e. 
RR*  ->  ( ( # `  { x  e.  dom  E  |  U  e.  ( E `  x ) } ) +e 0 )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) )
2825, 26, 273syl 20 . . . . . . . . . . . 12  |-  ( ( ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } )  =  0  /\  V USGrph  E
)  ->  ( ( # `
 { x  e. 
dom  E  |  U  e.  ( E `  x
) } ) +e 0 )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) )
2921, 28eqtrd 2498 . . . . . . . . . . 11  |-  ( ( ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } )  =  0  /\  V USGrph  E
)  ->  ( ( # `
 { x  e. 
dom  E  |  U  e.  ( E `  x
) } ) +e ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } ) )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) )
3029ex 434 . . . . . . . . . 10  |-  ( (
# `  { x  e.  dom  E  |  ( E `  x )  =  { U } } )  =  0  ->  ( V USGrph  E  ->  ( ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) +e
( # `  { x  e.  dom  E  |  ( E `  x )  =  { U } } ) )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) ) )
3119, 30syl 16 . . . . . . . . 9  |-  ( { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/)  ->  ( V USGrph  E  -> 
( ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) +e
( # `  { x  e.  dom  E  |  ( E `  x )  =  { U } } ) )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) ) )
3231com12 31 . . . . . . . 8  |-  ( V USGrph  E  ->  ( { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/)  ->  (
( # `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) +e ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } ) )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) ) )
3332adantl 466 . . . . . . 7  |-  ( ( ( ( V VDeg  E
) `  U )  =  ( ( # `  { x  e.  dom  E  |  U  e.  ( E `  x ) } ) +e
( # `  { x  e.  dom  E  |  ( E `  x )  =  { U } } ) )  /\  V USGrph  E )  ->  ( { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/)  ->  ( ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) +e
( # `  { x  e.  dom  E  |  ( E `  x )  =  { U } } ) )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) ) )
34 eqeq1 2461 . . . . . . . 8  |-  ( ( ( V VDeg  E ) `
 U )  =  ( ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) +e
( # `  { x  e.  dom  E  |  ( E `  x )  =  { U } } ) )  -> 
( ( ( V VDeg 
E ) `  U
)  =  ( # `  { x  e.  dom  E  |  U  e.  ( E `  x ) } )  <->  ( ( # `
 { x  e. 
dom  E  |  U  e.  ( E `  x
) } ) +e ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } ) )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) ) )
3534adantr 465 . . . . . . 7  |-  ( ( ( ( V VDeg  E
) `  U )  =  ( ( # `  { x  e.  dom  E  |  U  e.  ( E `  x ) } ) +e
( # `  { x  e.  dom  E  |  ( E `  x )  =  { U } } ) )  /\  V USGrph  E )  ->  (
( ( V VDeg  E
) `  U )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } )  <->  ( ( # `
 { x  e. 
dom  E  |  U  e.  ( E `  x
) } ) +e ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } ) )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) ) )
3633, 35sylibrd 234 . . . . . 6  |-  ( ( ( ( V VDeg  E
) `  U )  =  ( ( # `  { x  e.  dom  E  |  U  e.  ( E `  x ) } ) +e
( # `  { x  e.  dom  E  |  ( E `  x )  =  { U } } ) )  /\  V USGrph  E )  ->  ( { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/)  ->  ( ( V VDeg  E
) `  U )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) ) )
3736ex 434 . . . . 5  |-  ( ( ( V VDeg  E ) `
 U )  =  ( ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) +e
( # `  { x  e.  dom  E  |  ( E `  x )  =  { U } } ) )  -> 
( V USGrph  E  ->  ( { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/)  ->  ( ( V VDeg  E
) `  U )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) ) ) )
3815, 16, 373syl 20 . . . 4  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  ( V USGrph  E  ->  ( {
x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/)  ->  ( ( V VDeg  E
) `  U )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) ) ) )
3938com12 31 . . 3  |-  ( V USGrph  E  ->  ( ( V USGrph  E  /\  U  e.  V
)  ->  ( {
x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/)  ->  ( ( V VDeg  E
) `  U )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) ) ) )
4039anabsi5 817 . 2  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  ( { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/)  ->  ( ( V VDeg  E
) `  U )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) ) )
411, 40mpd 15 1  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  (
( V VDeg  E ) `  U )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109   (/)c0 3793   {csn 4032   class class class wbr 4456   dom cdm 5008   Fun wfun 5588    Fn wfn 5589   ` cfv 5594  (class class class)co 6296   0cc0 9509   RR*cxr 9644   +ecxad 11341   #chash 12408   USGrph cusg 24457   VDeg cvdg 25020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-xadd 11344  df-fz 11698  df-hash 12409  df-usgra 24460  df-vdgr 25021
This theorem is referenced by:  vdusgra0nedg  25035  hashnbgravd  25039  hashnbgravdg  25040  usgravd0nedg  25045  vdn0frgrav2  25150  vdgn0frgrav2  25151  vdn1frgrav2  25152  vdgn1frgrav2  25153
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