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Theorem umgra0 24001
Description: The empty graph, with vertices but no edges, is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
Assertion
Ref Expression
umgra0  |-  ( V  e.  W  ->  V UMGrph  (/) )

Proof of Theorem umgra0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 f0 5764 . . 3  |-  (/) : (/) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 }
2 dm0 5214 . . . 4  |-  dom  (/)  =  (/)
32feq2i 5722 . . 3  |-  ( (/) : dom  (/) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  <->  (/) :
(/) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
41, 3mpbir 209 . 2  |-  (/) : dom  (/) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 }
5 0ex 4577 . . 3  |-  (/)  e.  _V
6 isumgra 23991 . . 3  |-  ( ( V  e.  W  /\  (/) 
e.  _V )  ->  ( V UMGrph 
(/) 
<->  (/) : dom  (/) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
75, 6mpan2 671 . 2  |-  ( V  e.  W  ->  ( V UMGrph 
(/) 
<->  (/) : dom  (/) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
84, 7mpbiri 233 1  |-  ( V  e.  W  ->  V UMGrph  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1767   {crab 2818   _Vcvv 3113    \ cdif 3473   (/)c0 3785   ~Pcpw 4010   {csn 4027   class class class wbr 4447   dom cdm 4999   -->wf 5582   ` cfv 5586    <_ cle 9625   2c2 10581   #chash 12369   UMGrph cumg 23988
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-fun 5588  df-fn 5589  df-f 5590  df-umgra 23989
This theorem is referenced by:  eupa0  24650
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