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Theorem isuslgra 28113
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 5434 . . . 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 452 . . 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 4881 . . . . 5  |-  ( e  =  E  ->  dom  e  =  dom  E )
43adantl 452 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  dom  e  =  dom  E )
5 f1eq2 5435 . . . 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 15 . . 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 443 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
87pweqd 3632 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ~P v  =  ~P V )
98difeq1d 3295 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ~P v  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
10 rabeq 2784 . . . 4  |-  ( ( ~P v  \  { (/)
} )  =  ( ~P V  \  { (/)
} )  ->  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  <_  2 }  =  { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
11 f1eq3 5436 . . . 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 18 . . 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 270 . 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 28107 . 2  |- USLGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  <_  2 } }
1513, 14brabga 4281 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686   {crab 2549    \ cdif 3151   (/)c0 3457   ~Pcpw 3627   {csn 3642   class class class wbr 4025   dom cdm 4691   -1-1->wf1 5254   ` cfv 5257    <_ cle 8870   2c2 9797   #chash 11339   USLGrph cuslg 28105
This theorem is referenced by:  uslgraf  28115  uslisumgra  28123  usisuslgra  28124  uslgra1  28129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-uslgra 28107
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