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Theorem vdusgraval 23724
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 23446 . 2  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/) )
2 usgrafun 23424 . . . . . . 7  |-  ( V USGrph  E  ->  Fun  E )
3 usgrav 23417 . . . . . . 7  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
4 funfn 5550 . . . . . . . 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 6614 . . . . . . . . . . . 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 1168 . . . . . . . . 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 23713 . . . . 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 5794 . . . . . . . . . . 11  |-  ( { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/)  ->  ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } )  =  ( # `  (/) ) )
18 hash0 12247 . . . . . . . . . . 11  |-  ( # `  (/) )  =  0
1917, 18syl6eq 2509 . . . . . . . . . 10  |-  ( { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/)  ->  ( # `  {
x  e.  dom  E  |  ( E `  x )  =  { U } } )  =  0 )
20 oveq2 6203 . . . . . . . . . . . . 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 4545 . . . . . . . . . . . . . . 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 12239 . . . . . . . . . . . . 13  |-  ( { x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  _V  ->  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } )  e. 
RR* )
27 xaddid1 11315 . . . . . . . . . . . . 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 2493 . . . . . . . . . . 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 2456 . . . . . . . 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 813 . 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 965    = wceq 1370    e. wcel 1758   {crab 2800   _Vcvv 3072   (/)c0 3740   {csn 3980   class class class wbr 4395   dom cdm 4943   Fun wfun 5515    Fn wfn 5516   ` cfv 5521  (class class class)co 6195   0cc0 9388   RR*cxr 9523   +ecxad 11193   #chash 12215   USGrph cusg 23411   VDeg cvdg 23710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-n0 10686  df-z 10753  df-uz 10968  df-xadd 11196  df-fz 11550  df-hash 12216  df-usgra 23413  df-vdgr 23711
This theorem is referenced by:  vdusgra0nedg  23725  hashnbgravd  23727  hashnbgravdg  23728  usgravd0nedg  23729  vdn0frgrav2  30759  vdgn0frgrav2  30760  vdn1frgrav2  30761  vdgn1frgrav2  30762
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