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Theorem isusgra 24167
Description: The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Assertion
Ref Expression
isusgra  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
Distinct variable groups:    x, E    x, V
Allowed substitution hints:    W( x)    X( x)

Proof of Theorem isusgra
Dummy variables  v 
e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1eq1 5782 . . . 4  |-  ( e  =  E  ->  (
e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 }  <->  E : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } ) )
21adantl 466 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  ( e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 }  <->  E : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
3 dmeq 5209 . . . . 5  |-  ( e  =  E  ->  dom  e  =  dom  E )
43adantl 466 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  dom  e  =  dom  E )
5 f1eq2 5783 . . . 4  |-  ( dom  e  =  dom  E  ->  ( E : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 }  <->  E : dom  E -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
64, 5syl 16 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  ( E : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 }  <->  E : dom  E -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
7 simpl 457 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
87pweqd 4021 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ~P v  =  ~P V )
98difeq1d 3626 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ~P v  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
10 rabeq 3112 . . . 4  |-  ( ( ~P v  \  { (/)
} )  =  ( ~P V  \  { (/)
} )  ->  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }  =  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } )
11 f1eq3 5784 . . . 4  |-  ( { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 }  =  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 }  ->  ( E : dom  E -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 }  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
129, 10, 113syl 20 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  ( E : dom  E
-1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 }  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
132, 6, 123bitrd 279 . 2  |-  ( ( v  =  V  /\  e  =  E )  ->  ( e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 }  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
14 df-usgra 24156 . 2  |- USGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 } }
1513, 14brabga 4767 1  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2821    \ cdif 3478   (/)c0 3790   ~Pcpw 4016   {csn 4033   class class class wbr 4453   dom cdm 5005   -1-1->wf1 5591   ` cfv 5594   2c2 10597   #chash 12385   USGrph cusg 24153
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 4574  ax-nul 4582  ax-pr 4692
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-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-usgra 24156
This theorem is referenced by:  usgraf  24169  isusgra0  24170  usgraeq12d  24185  usisuslgra  24188  usgrares  24192  usgra0  24193  usgra0v  24194  usgra1v  24213
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