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Theorem usgra0 23434
Description: The empty graph, with vertices but no edges, is a graph, analogous to umgra0 23404. (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 5773 . . 3  |-  (/) : (/) -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }
2 dm0 5154 . . . 4  |-  dom  (/)  =  (/)
3 f1eq2 5703 . . . 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 4523 . . 3  |-  (/)  e.  _V
7 isusgra 23417 . . 3  |-  ( ( V  e.  W  /\  (/) 
e.  _V )  ->  ( V USGrph 
(/) 
<->  (/) : dom  (/) -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } ) )
86, 7mpan2 671 . 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 1370    e. wcel 1758   {crab 2799   _Vcvv 3071    \ cdif 3426   (/)c0 3738   ~Pcpw 3961   {csn 3978   class class class wbr 4393   dom cdm 4941   -1-1->wf1 5516   ` cfv 5519   2c2 10475   #chash 12213   USGrph cusg 23409
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-usgra 23411
This theorem is referenced by:  usgra0v  23435  usgra1v  23453  cusgra0v  23513  cusgra1v  23514
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