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Theorem isuslgra 23418
Description: The property of being an undirected simple graph with loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Assertion
Ref Expression
isuslgra  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USLGrph  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 isuslgra
Dummy variables  v 
e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1eq1 5704 . . . 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 5143 . . . . 5  |-  ( e  =  E  ->  dom  e  =  dom  E )
43adantl 466 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  dom  e  =  dom  E )
5 f1eq2 5705 . . . 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 3968 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ~P v  =  ~P V )
98difeq1d 3576 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ~P v  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
10 rabeq 3066 . . . 4  |-  ( ( ~P v  \  { (/)
} )  =  ( ~P V  \  { (/)
} )  ->  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  <_  2 }  =  { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
11 f1eq3 5706 . . . 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-uslgra 23412 . 2  |- USLGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  <_  2 } }
1513, 14brabga 4706 1  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USLGrph  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 1370    e. wcel 1758   {crab 2800    \ cdif 3428   (/)c0 3740   ~Pcpw 3963   {csn 3980   class class class wbr 4395   dom cdm 4943   -1-1->wf1 5518   ` cfv 5521    <_ cle 9525   2c2 10477   #chash 12215   USLGrph cuslg 23410
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 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-opab 4454  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-uslgra 23412
This theorem is referenced by:  uslgraf  23420  uslisumgra  23432  usisuslgra  23433  uslgra1  23438
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