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Theorem usgra0 24572
Description: The empty graph, with vertices but no edges, is a graph, analogous to umgra0 24527. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Assertion
Ref Expression
usgra0  |-  ( V  e.  W  ->  V USGrph  (/) )

Proof of Theorem usgra0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 f10 5829 . . 3  |-  (/) : (/) -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }
2 dm0 5205 . . . 4  |-  dom  (/)  =  (/)
3 f1eq2 5759 . . . 4  |-  ( dom  (/)  =  (/)  ->  ( (/) : dom  (/) -1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 }  <->  (/) : (/) -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 } ) )
42, 3ax-mp 5 . . 3  |-  ( (/) : dom  (/) -1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 }  <->  (/) : (/) -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 } )
51, 4mpbir 209 . 2  |-  (/) : dom  (/) -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }
6 0ex 4569 . . 3  |-  (/)  e.  _V
7 isusgra 24546 . . 3  |-  ( ( V  e.  W  /\  (/) 
e.  _V )  ->  ( V USGrph 
(/) 
<->  (/) : dom  (/) -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } ) )
86, 7mpan2 669 . 2  |-  ( V  e.  W  ->  ( V USGrph 
(/) 
<->  (/) : dom  (/) -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } ) )
95, 8mpbiri 233 1  |-  ( V  e.  W  ->  V USGrph  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    e. wcel 1823   {crab 2808   _Vcvv 3106    \ cdif 3458   (/)c0 3783   ~Pcpw 3999   {csn 4016   class class class wbr 4439   dom cdm 4988   -1-1->wf1 5567   ` cfv 5570   2c2 10581   #chash 12387   USGrph cusg 24532
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-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  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-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-usgra 24535
This theorem is referenced by:  usgra0v  24573  usgra1v  24592  cusgra0v  24662  cusgra1v  24663
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