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Theorem umgra0 21228
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 5568 . . 3  |-  (/) : (/) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 }
2 dm0 5024 . . . 4  |-  dom  (/)  =  (/)
32feq2i 5527 . . 3  |-  ( (/) : dom  (/) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  <->  (/) :
(/) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
41, 3mpbir 201 . 2  |-  (/) : dom  (/) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 }
5 0ex 4281 . . 3  |-  (/)  e.  _V
6 isumgra 21218 . . 3  |-  ( ( V  e.  W  /\  (/) 
e.  _V )  ->  ( V UMGrph 
(/) 
<->  (/) : dom  (/) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
75, 6mpan2 653 . 2  |-  ( V  e.  W  ->  ( V UMGrph 
(/) 
<->  (/) : dom  (/) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
84, 7mpbiri 225 1  |-  ( V  e.  W  ->  V UMGrph  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1717   {crab 2654   _Vcvv 2900    \ cdif 3261   (/)c0 3572   ~Pcpw 3743   {csn 3758   class class class wbr 4154   dom cdm 4819   -->wf 5391   ` cfv 5395    <_ cle 9055   2c2 9982   #chash 11546   UMGrph cumg 21215
This theorem is referenced by:  eupa0  21545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-fun 5397  df-fn 5398  df-f 5399  df-umgra 21216
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