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Theorem isuslgra 24470
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 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 5213 . . . . 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 4020 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ~P v  =  ~P V )
98difeq1d 3617 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ~P v  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
10 rabeq 3103 . . . 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-uslgra 24459 . 2  |- USLGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  <_  2 } }
1513, 14brabga 4770 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 1395    e. wcel 1819   {crab 2811    \ cdif 3468   (/)c0 3793   ~Pcpw 4015   {csn 4032   class class class wbr 4456   dom cdm 5008   -1-1->wf1 5591   ` cfv 5594    <_ cle 9646   2c2 10606   #chash 12408   USLGrph cuslg 24456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-uslgra 24459
This theorem is referenced by:  uslgraf  24472  uslisushgra  24490  uslisumgra  24491  usisuslgra  24492  uslgra1  24499
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