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Theorem uhg0e 32748
Description: The empty graph, with vertices but no edges, is a hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 18-Jan-2020.)
Assertion
Ref Expression
uhg0e  |-  ( ( G  e.  W  /\  ( .ef  `  G )  =  (/) )  ->  G  e. UHGraph  )

Proof of Theorem uhg0e
StepHypRef Expression
1 f0 5748 . . 3  |-  (/) : (/) --> ( ~P ( Base `  G
)  \  { (/) } )
2 dm0 5205 . . . 4  |-  dom  (/)  =  (/)
32feq2i 5706 . . 3  |-  ( (/) : dom  (/) --> ( ~P ( Base `  G )  \  { (/) } )  <->  (/) : (/) --> ( ~P ( Base `  G
)  \  { (/) } ) )
41, 3mpbir 209 . 2  |-  (/) : dom  (/) --> ( ~P ( Base `  G
)  \  { (/) } )
5 eqid 2454 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
6 eqid 2454 . . . 4  |-  ( .ef  `  G )  =  ( .ef  `  G )
75, 6isuhgr 32738 . . 3  |-  ( G  e.  W  ->  ( G  e. UHGraph  <->  ( .ef  `  G ) : dom  ( .ef  `  G ) --> ( ~P ( Base `  G
)  \  { (/) } ) ) )
8 id 22 . . . 4  |-  ( ( .ef  `  G )  =  (/)  ->  ( .ef  `  G )  =  (/) )
9 dmeq 5192 . . . 4  |-  ( ( .ef  `  G )  =  (/)  ->  dom  ( .ef  `  G
)  =  dom  (/) )
108, 9feq12d 5702 . . 3  |-  ( ( .ef  `  G )  =  (/)  ->  ( ( .ef  `  G
) : dom  ( .ef  `  G ) --> ( ~P ( Base `  G
)  \  { (/) } )  <->  (/)
: dom  (/) --> ( ~P ( Base `  G
)  \  { (/) } ) ) )
117, 10sylan9bb 697 . 2  |-  ( ( G  e.  W  /\  ( .ef  `  G )  =  (/) )  ->  ( G  e. UHGraph 
<->  (/) : dom  (/) --> ( ~P ( Base `  G
)  \  { (/) } ) ) )
124, 11mpbiri 233 1  |-  ( ( G  e.  W  /\  ( .ef  `  G )  =  (/) )  ->  G  e. UHGraph  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    \ cdif 3458   (/)c0 3783   ~Pcpw 3999   {csn 4016   dom cdm 4988   -->wf 5566   ` cfv 5570   Basecbs 14716   .ef cedgf 32732   UHGraph cuhgr 32733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-uhgr 32736
This theorem is referenced by:  uhg0v  32749
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