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Theorem hashnbgravdg 25115
Description: The size of the set of the neighbors of a vertex is the vertex degree of this vertex, analogous to hashnbgravd 25114. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
Assertion
Ref Expression
hashnbgravdg  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  U ) )  =  ( ( V VDeg  E
) `  U )
)

Proof of Theorem hashnbgravdg
Dummy variables  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nbgraf1o 24649 . . . 4  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  E. f 
f : ( <. V ,  E >. Neighbors  U
)
-1-1-onto-> { x  e.  dom  E  |  U  e.  ( E `  x ) } )
2 hasheqf1o 12404 . . . 4  |-  ( ( ( <. V ,  E >. Neighbors  U )  e.  Fin  /\ 
{ x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  Fin )  ->  ( ( # `  ( <. V ,  E >. Neighbors  U
) )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } )  <->  E. f 
f : ( <. V ,  E >. Neighbors  U
)
-1-1-onto-> { x  e.  dom  E  |  U  e.  ( E `  x ) } ) )
31, 2syl5ibr 221 . . 3  |-  ( ( ( <. V ,  E >. Neighbors  U )  e.  Fin  /\ 
{ x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  Fin )  ->  ( ( V USGrph  E  /\  U  e.  V
)  ->  ( # `  ( <. V ,  E >. Neighbors  U
) )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) ) )
4 edgusgranbfin 24652 . . . . . 6  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  (
( <. V ,  E >. Neighbors  U )  e.  Fin  <->  {
x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  Fin )
)
5 pm2.24 109 . . . . . 6  |-  ( { x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  Fin  ->  ( -.  { x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  Fin  ->  (
# `  ( <. V ,  E >. Neighbors  U ) )  =  ( # `  { x  e.  dom  E  |  U  e.  ( E `  x ) } ) ) )
64, 5syl6bi 228 . . . . 5  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  (
( <. V ,  E >. Neighbors  U )  e.  Fin  ->  ( -.  { x  e.  dom  E  |  U  e.  ( E `  x
) }  e.  Fin  ->  ( # `  ( <. V ,  E >. Neighbors  U
) )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) ) ) )
76com3l 81 . . . 4  |-  ( (
<. V ,  E >. Neighbors  U
)  e.  Fin  ->  ( -.  { x  e. 
dom  E  |  U  e.  ( E `  x
) }  e.  Fin  ->  ( ( V USGrph  E  /\  U  e.  V
)  ->  ( # `  ( <. V ,  E >. Neighbors  U
) )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) ) ) )
87imp 427 . . 3  |-  ( ( ( <. V ,  E >. Neighbors  U )  e.  Fin  /\ 
-.  { x  e. 
dom  E  |  U  e.  ( E `  x
) }  e.  Fin )  ->  ( ( V USGrph  E  /\  U  e.  V
)  ->  ( # `  ( <. V ,  E >. Neighbors  U
) )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) ) )
9 pm2.24 109 . . . . . 6  |-  ( (
<. V ,  E >. Neighbors  U
)  e.  Fin  ->  ( -.  ( <. V ,  E >. Neighbors  U )  e.  Fin  ->  ( # `  ( <. V ,  E >. Neighbors  U
) )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) ) )
104, 9syl6bir 229 . . . . 5  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  ( { x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  Fin  ->  ( -.  ( <. V ,  E >. Neighbors  U )  e.  Fin  ->  ( # `  ( <. V ,  E >. Neighbors  U
) )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) ) ) )
1110com13 80 . . . 4  |-  ( -.  ( <. V ,  E >. Neighbors  U )  e.  Fin  ->  ( { x  e. 
dom  E  |  U  e.  ( E `  x
) }  e.  Fin  ->  ( ( V USGrph  E  /\  U  e.  V
)  ->  ( # `  ( <. V ,  E >. Neighbors  U
) )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) ) ) )
1211imp 427 . . 3  |-  ( ( -.  ( <. V ,  E >. Neighbors  U )  e.  Fin  /\ 
{ x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  Fin )  ->  ( ( V USGrph  E  /\  U  e.  V
)  ->  ( # `  ( <. V ,  E >. Neighbors  U
) )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) ) )
13 ovex 6298 . . . . . . 7  |-  ( <. V ,  E >. Neighbors  U
)  e.  _V
1413a1i 11 . . . . . 6  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  ( <. V ,  E >. Neighbors  U
)  e.  _V )
15 simpl 455 . . . . . 6  |-  ( ( -.  ( <. V ,  E >. Neighbors  U )  e.  Fin  /\ 
-.  { x  e. 
dom  E  |  U  e.  ( E `  x
) }  e.  Fin )  ->  -.  ( <. V ,  E >. Neighbors  U )  e.  Fin )
16 hashinf 12392 . . . . . 6  |-  ( ( ( <. V ,  E >. Neighbors  U )  e.  _V  /\ 
-.  ( <. V ,  E >. Neighbors  U )  e.  Fin )  ->  ( # `  ( <. V ,  E >. Neighbors  U
) )  = +oo )
1714, 15, 16syl2anr 476 . . . . 5  |-  ( ( ( -.  ( <. V ,  E >. Neighbors  U
)  e.  Fin  /\  -.  { x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  Fin )  /\  ( V USGrph  E  /\  U  e.  V )
)  ->  ( # `  ( <. V ,  E >. Neighbors  U
) )  = +oo )
18 usgrav 24540 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
1918simprd 461 . . . . . . . 8  |-  ( V USGrph  E  ->  E  e.  _V )
20 dmexg 6704 . . . . . . . 8  |-  ( E  e.  _V  ->  dom  E  e.  _V )
21 rabexg 4587 . . . . . . . 8  |-  ( dom 
E  e.  _V  ->  { x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  _V )
2219, 20, 213syl 20 . . . . . . 7  |-  ( V USGrph  E  ->  { x  e. 
dom  E  |  U  e.  ( E `  x
) }  e.  _V )
2322adantr 463 . . . . . 6  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  { x  e.  dom  E  |  U  e.  ( E `  x
) }  e.  _V )
24 simpr 459 . . . . . 6  |-  ( ( -.  ( <. V ,  E >. Neighbors  U )  e.  Fin  /\ 
-.  { x  e. 
dom  E  |  U  e.  ( E `  x
) }  e.  Fin )  ->  -.  { x  e.  dom  E  |  U  e.  ( E `  x
) }  e.  Fin )
25 hashinf 12392 . . . . . 6  |-  ( ( { x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  _V  /\  -.  { x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  Fin )  ->  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } )  = +oo )
2623, 24, 25syl2anr 476 . . . . 5  |-  ( ( ( -.  ( <. V ,  E >. Neighbors  U
)  e.  Fin  /\  -.  { x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  Fin )  /\  ( V USGrph  E  /\  U  e.  V )
)  ->  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } )  = +oo )
2717, 26eqtr4d 2498 . . . 4  |-  ( ( ( -.  ( <. V ,  E >. Neighbors  U
)  e.  Fin  /\  -.  { x  e.  dom  E  |  U  e.  ( E `  x ) }  e.  Fin )  /\  ( V USGrph  E  /\  U  e.  V )
)  ->  ( # `  ( <. V ,  E >. Neighbors  U
) )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) )
2827ex 432 . . 3  |-  ( ( -.  ( <. V ,  E >. Neighbors  U )  e.  Fin  /\ 
-.  { x  e. 
dom  E  |  U  e.  ( E `  x
) }  e.  Fin )  ->  ( ( V USGrph  E  /\  U  e.  V
)  ->  ( # `  ( <. V ,  E >. Neighbors  U
) )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) ) )
293, 8, 12, 284cases 947 . 2  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  U ) )  =  ( # `  {
x  e.  dom  E  |  U  e.  ( E `  x ) } ) )
30 vdusgraval 25109 . 2  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  (
( V VDeg  E ) `  U )  =  (
# `  { x  e.  dom  E  |  U  e.  ( E `  x
) } ) )
3129, 30eqtr4d 2498 1  |-  ( ( V USGrph  E  /\  U  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  U ) )  =  ( ( V VDeg  E
) `  U )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   {crab 2808   _Vcvv 3106   <.cop 4022   class class class wbr 4439   dom cdm 4988   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270   Fincfn 7509   +oocpnf 9614   #chash 12387   USGrph cusg 24532   Neighbors cnbgra 24619   VDeg cvdg 25095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-xadd 11322  df-fz 11676  df-hash 12388  df-usgra 24535  df-nbgra 24622  df-vdgr 25096
This theorem is referenced by:  nbhashnn0  25116  usgrauvtxvdbi  25122  cusgraisrusgra  25140  rusgraprop2  25144  frgrancvvdgeq  25245  usgreghash2spotv  25268  vdusgravaledg  32729  usgrauvtxvd  32730  vdcusgra  32731
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