MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isusgra Structured version   Unicode version

Theorem isusgra 23393
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 5685 . . . 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 5124 . . . . 5  |-  ( e  =  E  ->  dom  e  =  dom  E )
43adantl 466 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  dom  e  =  dom  E )
5 f1eq2 5686 . . . 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 3949 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ~P v  =  ~P V )
98difeq1d 3557 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ~P v  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
10 rabeq 3048 . . . 4  |-  ( ( ~P v  \  { (/)
} )  =  ( ~P V  \  { (/)
} )  ->  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }  =  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 } )
11 f1eq3 5687 . . . 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 23387 . 2  |- USGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 } }
1513, 14brabga 4687 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 1370    e. wcel 1757   {crab 2796    \ cdif 3409   (/)c0 3721   ~Pcpw 3944   {csn 3961   class class class wbr 4376   dom cdm 4924   -1-1->wf1 5499   ` cfv 5502   2c2 10458   #chash 12190   USGrph cusg 23385
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 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615
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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-rab 2801  df-v 3056  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-br 4377  df-opab 4435  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-usgra 23387
This theorem is referenced by:  usgraf  23395  isusgra0  23396  usgraeq12d  23405  usisuslgra  23407  usgrares  23409  usgra0  23410  usgra0v  23411  usgra1v  23429
  Copyright terms: Public domain W3C validator