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Theorem isuhgra 24597
Description: The property of being an undirected hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
Assertion
Ref Expression
isuhgra  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V UHGrph  E  <->  E : dom  E --> ( ~P V  \  { (/) } ) ) )

Proof of Theorem isuhgra
Dummy variables  v 
e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 459 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  e  =  E )
2 dmeq 5143 . . . 4  |-  ( e  =  E  ->  dom  e  =  dom  E )
32adantl 464 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  dom  e  =  dom  E )
4 pweq 3955 . . . . 5  |-  ( v  =  V  ->  ~P v  =  ~P V
)
54difeq1d 3557 . . . 4  |-  ( v  =  V  ->  ( ~P v  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
65adantr 463 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ~P v  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
71, 3, 6feq123d 5658 . 2  |-  ( ( v  =  V  /\  e  =  E )  ->  ( e : dom  e
--> ( ~P v  \  { (/) } )  <->  E : dom  E --> ( ~P V  \  { (/) } ) ) )
8 df-uhgra 24591 . 2  |- UHGrph  =  { <. v ,  e >.  |  e : dom  e
--> ( ~P v  \  { (/) } ) }
97, 8brabga 4701 1  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V UHGrph  E  <->  E : dom  E --> ( ~P V  \  { (/) } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1403    e. wcel 1840    \ cdif 3408   (/)c0 3735   ~Pcpw 3952   {csn 3969   class class class wbr 4392   dom cdm 4940   -->wf 5519   UHGrph cuhg 24589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-br 4393  df-opab 4451  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-fun 5525  df-fn 5526  df-f 5527  df-uhgra 24591
This theorem is referenced by:  uhgraf  24598  ushgrauhgra  24602  uhgraop  24603  uhgraeq12d  24606  uhgrares  24607  uhgra0  24608  uhgra0v  24609  uhgraun  24610  umisuhgra  24626  uhguhgra  37934
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