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Theorem isuhgr 32671
Description: The predicate "is an undirected hypergraph." (Contributed by AV, 18-Jan-2020.)
Hypotheses
Ref Expression
isuhgr.v  |-  V  =  ( Base `  G
)
isuhgr.e  |-  E  =  ( .ef  `  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 32669 . . 3  |- UHGraph  =  {
g  |  [. ( Base `  g )  / 
v ]. [. ( .ef  `  g )  /  e ]. e : dom  e --> ( ~P v  \  { (/)
} ) }
21eleq2i 2535 . 2  |-  ( G  e. UHGraph 
<->  G  e.  { g  |  [. ( Base `  g )  /  v ]. [. ( .ef  `  g
)  /  e ]. e : dom  e --> ( ~P v  \  { (/)
} ) } )
3 fveq2 5872 . . . . 5  |-  ( h  =  G  ->  ( .ef  `  h )  =  ( .ef  `  G ) )
4 isuhgr.e . . . . 5  |-  E  =  ( .ef  `  G )
53, 4syl6eqr 2516 . . . 4  |-  ( h  =  G  ->  ( .ef  `  h )  =  E )
63dmeqd 5215 . . . . 5  |-  ( h  =  G  ->  dom  ( .ef  `  h )  =  dom  ( .ef  `  G
) )
74eqcomi 2470 . . . . . 6  |-  ( .ef  `  G )  =  E
87dmeqi 5214 . . . . 5  |-  dom  ( .ef  `  G )  =  dom  E
96, 8syl6eq 2514 . . . 4  |-  ( h  =  G  ->  dom  ( .ef  `  h )  =  dom  E )
10 fveq2 5872 . . . . . . 7  |-  ( h  =  G  ->  ( Base `  h )  =  ( Base `  G
) )
11 isuhgr.v . . . . . . 7  |-  V  =  ( Base `  G
)
1210, 11syl6eqr 2516 . . . . . 6  |-  ( h  =  G  ->  ( Base `  h )  =  V )
1312pweqd 4020 . . . . 5  |-  ( h  =  G  ->  ~P ( Base `  h )  =  ~P V )
1413difeq1d 3617 . . . 4  |-  ( h  =  G  ->  ( ~P ( Base `  h
)  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
155, 9, 14feq123d 5727 . . 3  |-  ( h  =  G  ->  (
( .ef  `  h ) : dom  ( .ef  `  h
) --> ( ~P ( Base `  h )  \  { (/) } )  <->  E : dom  E --> ( ~P V  \  { (/) } ) ) )
16 fvex 5882 . . . . . 6  |-  ( Base `  g )  e.  _V
1716a1i 11 . . . . 5  |-  ( g  =  h  ->  ( Base `  g )  e. 
_V )
18 fveq2 5872 . . . . 5  |-  ( g  =  h  ->  ( Base `  g )  =  ( Base `  h
) )
19 fvex 5882 . . . . . . 7  |-  ( .ef  `  g )  e.  _V
2019a1i 11 . . . . . 6  |-  ( ( g  =  h  /\  v  =  ( Base `  h ) )  -> 
( .ef  `  g )  e.  _V )
21 fveq2 5872 . . . . . . 7  |-  ( g  =  h  ->  ( .ef  `  g )  =  ( .ef  `  h ) )
2221adantr 465 . . . . . 6  |-  ( ( g  =  h  /\  v  =  ( Base `  h ) )  -> 
( .ef  `  g )  =  ( .ef  `  h ) )
23 simpr 461 . . . . . . 7  |-  ( ( ( g  =  h  /\  v  =  (
Base `  h )
)  /\  e  =  ( .ef  `  h ) )  ->  e  =  ( .ef  `  h ) )
2423dmeqd 5215 . . . . . . 7  |-  ( ( ( g  =  h  /\  v  =  (
Base `  h )
)  /\  e  =  ( .ef  `  h ) )  ->  dom  e  =  dom  ( .ef  `  h ) )
25 simpr 461 . . . . . . . . . 10  |-  ( ( g  =  h  /\  v  =  ( Base `  h ) )  -> 
v  =  ( Base `  h ) )
2625pweqd 4020 . . . . . . . . 9  |-  ( ( g  =  h  /\  v  =  ( Base `  h ) )  ->  ~P v  =  ~P ( Base `  h )
)
2726difeq1d 3617 . . . . . . . 8  |-  ( ( g  =  h  /\  v  =  ( Base `  h ) )  -> 
( ~P v  \  { (/) } )  =  ( ~P ( Base `  h )  \  { (/)
} ) )
2827adantr 465 . . . . . . 7  |-  ( ( ( g  =  h  /\  v  =  (
Base `  h )
)  /\  e  =  ( .ef  `  h ) )  ->  ( ~P v  \  { (/) } )  =  ( ~P ( Base `  h )  \  { (/)
} ) )
2923, 24, 28feq123d 5727 . . . . . 6  |-  ( ( ( g  =  h  /\  v  =  (
Base `  h )
)  /\  e  =  ( .ef  `  h ) )  ->  ( e : dom  e --> ( ~P v  \  { (/) } )  <->  ( .ef  `  h ) : dom  ( .ef  `  h ) --> ( ~P ( Base `  h
)  \  { (/) } ) ) )
3020, 22, 29sbcied2 3365 . . . . 5  |-  ( ( g  =  h  /\  v  =  ( Base `  h ) )  -> 
( [. ( .ef  `  g
)  /  e ]. e : dom  e --> ( ~P v  \  { (/)
} )  <->  ( .ef  `  h
) : dom  ( .ef  `  h ) --> ( ~P ( Base `  h
)  \  { (/) } ) ) )
3117, 18, 30sbcied2 3365 . . . 4  |-  ( g  =  h  ->  ( [. ( Base `  g
)  /  v ]. [. ( .ef  `  g )  /  e ]. e : dom  e --> ( ~P v  \  { (/) } )  <->  ( .ef  `  h ) : dom  ( .ef  `  h ) --> ( ~P ( Base `  h
)  \  { (/) } ) ) )
3231cbvabv 2600 . . 3  |-  { g  |  [. ( Base `  g )  /  v ]. [. ( .ef  `  g
)  /  e ]. e : dom  e --> ( ~P v  \  { (/)
} ) }  =  { h  |  ( .ef  `  h ) : dom  ( .ef  `  h ) --> ( ~P ( Base `  h
)  \  { (/) } ) }
3315, 32elab2g 3248 . 2  |-  ( G  e.  U  ->  ( G  e.  { g  |  [. ( Base `  g
)  /  v ]. [. ( .ef  `  g )  /  e ]. e : dom  e --> ( ~P v  \  { (/) } ) }  <->  E : dom  E --> ( ~P V  \  { (/) } ) ) )
342, 33syl5bb 257 1  |-  ( G  e.  U  ->  ( G  e. UHGraph  <->  E : dom  E --> ( ~P V  \  { (/)
} ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   _Vcvv 3109   [.wsbc 3327    \ cdif 3468   (/)c0 3793   ~Pcpw 4015   {csn 4032   dom cdm 5008   -->wf 5590   ` cfv 5594   Basecbs 14735   .ef cedgf 32665   UHGraph cuhgr 32666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-uhgr 32669
This theorem is referenced by:  uhgf  32673  uhgeq12g  32675  ushguhg  32676  uhgrauhg  32678  uhg0e  32681  uhgrepe  32683
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