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Theorem isuhgra 23966
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 461 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  e  =  E )
2 dmeq 5196 . . . 4  |-  ( e  =  E  ->  dom  e  =  dom  E )
32adantl 466 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  dom  e  =  dom  E )
4 pweq 4008 . . . . 5  |-  ( v  =  V  ->  ~P v  =  ~P V
)
54difeq1d 3616 . . . 4  |-  ( v  =  V  ->  ( ~P v  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
65adantr 465 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ~P v  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
71, 3, 6feq123d 5714 . 2  |-  ( ( v  =  V  /\  e  =  E )  ->  ( e : dom  e
--> ( ~P v  \  { (/) } )  <->  E : dom  E --> ( ~P V  \  { (/) } ) ) )
8 df-uhgra 23962 . 2  |- UHGrph  =  { <. v ,  e >.  |  e : dom  e
--> ( ~P v  \  { (/) } ) }
97, 8brabga 4756 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 369    = wceq 1374    e. wcel 1762    \ cdif 3468   (/)c0 3780   ~Pcpw 4005   {csn 4022   class class class wbr 4442   dom cdm 4994   -->wf 5577   UHGrph cuhg 23961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-fun 5583  df-fn 5584  df-f 5585  df-uhgra 23962
This theorem is referenced by:  uhgraf  23967  uhgraop  23969  uhgraeq12d  23972  uhgrares  23973  uhgra0  23974  uhgra0v  23975  uhgraun  23976  umisuhgra  23992
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