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Theorem vdusgraval 24721
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 24194 . 2  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/) )
2 usgrafun 24163 . . . . . . 7  |-  ( V USGrph  E  ->  Fun  E )
3 usgrav 24152 . . . . . . 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 6726 . . . . . . . . . . . 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 24710 . . . . 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 12417 . . . . . . . . . . 11  |-  ( # `  (/) )  =  0
1917, 18syl6eq 2524 . . . . . . . . . 10  |-  ( { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/)  ->  ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } )  =  0 )
20 oveq2 6303 . . . . . . . . . . . . 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 4603 . . . . . . . . . . . . . . 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 12409 . . . . . . . . . . . . 13  |-  ( { x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  _V  ->  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } )  e. 
RR* )
27 xaddid1 11450 . . . . . . . . . . . . 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 2508 . . . . . . . . . . 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 2471 . . . . . . . 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 815 . 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 1379    e. wcel 1767   {crab 2821   _Vcvv 3118   (/)c0 3790   {csn 4033   class class class wbr 4453   dom cdm 5005   Fun wfun 5588    Fn wfn 5589   ` cfv 5594  (class class class)co 6295   0cc0 9504   RR*cxr 9639   +ecxad 11328   #chash 12385   USGrph cusg 24144   VDeg cvdg 24707
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-8 1769  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-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  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-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  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-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-xadd 11331  df-fz 11685  df-hash 12386  df-usgra 24147  df-vdgr 24708
This theorem is referenced by:  vdusgra0nedg  24722  hashnbgravd  24726  hashnbgravdg  24727  usgravd0nedg  24732  vdn0frgrav2  24838  vdgn0frgrav2  24839  vdn1frgrav2  24840  vdgn1frgrav2  24841
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