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Theorem isuhgr 39161
Description: The predicate "is an undirected hypergraph." (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.)
Hypotheses
Ref Expression
isuhgr.v  |-  V  =  (Vtx `  G )
isuhgr.e  |-  E  =  (iEdg `  G )
Assertion
Ref Expression
isuhgr  |-  ( G  e.  U  ->  ( G  e. UHGraph  <->  E : dom  E --> ( ~P V  \  { (/)
} ) ) )

Proof of Theorem isuhgr
Dummy variables  g  h  v  e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-uhgr 39159 . . 3  |- UHGraph  =  {
g  |  [. (Vtx `  g )  /  v ]. [. (iEdg `  g
)  /  e ]. e : dom  e --> ( ~P v  \  { (/)
} ) }
21eleq2i 2523 . 2  |-  ( G  e. UHGraph 
<->  G  e.  { g  |  [. (Vtx `  g )  /  v ]. [. (iEdg `  g
)  /  e ]. e : dom  e --> ( ~P v  \  { (/)
} ) } )
3 fveq2 5870 . . . . 5  |-  ( h  =  G  ->  (iEdg `  h )  =  (iEdg `  G ) )
4 isuhgr.e . . . . 5  |-  E  =  (iEdg `  G )
53, 4syl6eqr 2505 . . . 4  |-  ( h  =  G  ->  (iEdg `  h )  =  E )
63dmeqd 5040 . . . . 5  |-  ( h  =  G  ->  dom  (iEdg `  h )  =  dom  (iEdg `  G
) )
74eqcomi 2462 . . . . . 6  |-  (iEdg `  G )  =  E
87dmeqi 5039 . . . . 5  |-  dom  (iEdg `  G )  =  dom  E
96, 8syl6eq 2503 . . . 4  |-  ( h  =  G  ->  dom  (iEdg `  h )  =  dom  E )
10 fveq2 5870 . . . . . . 7  |-  ( h  =  G  ->  (Vtx `  h )  =  (Vtx
`  G ) )
11 isuhgr.v . . . . . . 7  |-  V  =  (Vtx `  G )
1210, 11syl6eqr 2505 . . . . . 6  |-  ( h  =  G  ->  (Vtx `  h )  =  V )
1312pweqd 3958 . . . . 5  |-  ( h  =  G  ->  ~P (Vtx `  h )  =  ~P V )
1413difeq1d 3552 . . . 4  |-  ( h  =  G  ->  ( ~P (Vtx `  h )  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
155, 9, 14feq123d 5723 . . 3  |-  ( h  =  G  ->  (
(iEdg `  h ) : dom  (iEdg `  h
) --> ( ~P (Vtx `  h )  \  { (/)
} )  <->  E : dom  E --> ( ~P V  \  { (/) } ) ) )
16 fvex 5880 . . . . . 6  |-  (Vtx `  g )  e.  _V
1716a1i 11 . . . . 5  |-  ( g  =  h  ->  (Vtx `  g )  e.  _V )
18 fveq2 5870 . . . . 5  |-  ( g  =  h  ->  (Vtx `  g )  =  (Vtx
`  h ) )
19 fvex 5880 . . . . . . 7  |-  (iEdg `  g )  e.  _V
2019a1i 11 . . . . . 6  |-  ( ( g  =  h  /\  v  =  (Vtx `  h
) )  ->  (iEdg `  g )  e.  _V )
21 fveq2 5870 . . . . . . 7  |-  ( g  =  h  ->  (iEdg `  g )  =  (iEdg `  h ) )
2221adantr 467 . . . . . 6  |-  ( ( g  =  h  /\  v  =  (Vtx `  h
) )  ->  (iEdg `  g )  =  (iEdg `  h ) )
23 simpr 463 . . . . . . 7  |-  ( ( ( g  =  h  /\  v  =  (Vtx
`  h ) )  /\  e  =  (iEdg `  h ) )  -> 
e  =  (iEdg `  h ) )
2423dmeqd 5040 . . . . . . 7  |-  ( ( ( g  =  h  /\  v  =  (Vtx
`  h ) )  /\  e  =  (iEdg `  h ) )  ->  dom  e  =  dom  (iEdg `  h ) )
25 simpr 463 . . . . . . . . . 10  |-  ( ( g  =  h  /\  v  =  (Vtx `  h
) )  ->  v  =  (Vtx `  h )
)
2625pweqd 3958 . . . . . . . . 9  |-  ( ( g  =  h  /\  v  =  (Vtx `  h
) )  ->  ~P v  =  ~P (Vtx `  h ) )
2726difeq1d 3552 . . . . . . . 8  |-  ( ( g  =  h  /\  v  =  (Vtx `  h
) )  ->  ( ~P v  \  { (/) } )  =  ( ~P (Vtx `  h )  \  { (/) } ) )
2827adantr 467 . . . . . . 7  |-  ( ( ( g  =  h  /\  v  =  (Vtx
`  h ) )  /\  e  =  (iEdg `  h ) )  -> 
( ~P v  \  { (/) } )  =  ( ~P (Vtx `  h )  \  { (/)
} ) )
2923, 24, 28feq123d 5723 . . . . . 6  |-  ( ( ( g  =  h  /\  v  =  (Vtx
`  h ) )  /\  e  =  (iEdg `  h ) )  -> 
( e : dom  e
--> ( ~P v  \  { (/) } )  <->  (iEdg `  h
) : dom  (iEdg `  h ) --> ( ~P (Vtx `  h )  \  { (/) } ) ) )
3020, 22, 29sbcied2 3307 . . . . 5  |-  ( ( g  =  h  /\  v  =  (Vtx `  h
) )  ->  ( [. (iEdg `  g )  /  e ]. e : dom  e --> ( ~P v  \  { (/) } )  <->  (iEdg `  h ) : dom  (iEdg `  h
) --> ( ~P (Vtx `  h )  \  { (/)
} ) ) )
3117, 18, 30sbcied2 3307 . . . 4  |-  ( g  =  h  ->  ( [. (Vtx `  g )  /  v ]. [. (iEdg `  g )  /  e ]. e : dom  e --> ( ~P v  \  { (/)
} )  <->  (iEdg `  h
) : dom  (iEdg `  h ) --> ( ~P (Vtx `  h )  \  { (/) } ) ) )
3231cbvabv 2577 . . 3  |-  { g  |  [. (Vtx `  g )  /  v ]. [. (iEdg `  g
)  /  e ]. e : dom  e --> ( ~P v  \  { (/)
} ) }  =  { h  |  (iEdg `  h ) : dom  (iEdg `  h ) --> ( ~P (Vtx `  h
)  \  { (/) } ) }
3315, 32elab2g 3189 . 2  |-  ( G  e.  U  ->  ( G  e.  { g  |  [. (Vtx `  g
)  /  v ]. [. (iEdg `  g )  /  e ]. e : dom  e --> ( ~P v  \  { (/) } ) }  <->  E : dom  E --> ( ~P V  \  { (/) } ) ) )
342, 33syl5bb 261 1  |-  ( G  e.  U  ->  ( G  e. UHGraph  <->  E : dom  E --> ( ~P V  \  { (/)
} ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1446    e. wcel 1889   {cab 2439   _Vcvv 3047   [.wsbc 3269    \ cdif 3403   (/)c0 3733   ~Pcpw 3953   {csn 3970   dom cdm 4837   -->wf 5581   ` cfv 5585  Vtxcvtx 39111  iEdgciedg 39112   UHGraph cuhgr 39157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-nul 4537
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-fv 5593  df-uhgr 39159
This theorem is referenced by:  uhgrf  39163  uhgreq12g  39166  ushgruhgr  39169  uhgrauhgr  39171  isuhgrop  39173  uhgr0e  39174  uhgr0  39176  uhgrun  39177  incistruhgr  39181  upgruhgr  39201  subuhgr  39368
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