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Theorem uhgraun 23392
Description: If  <. V ,  E >. and  <. V ,  F >. are hypergraphs, then  <. V ,  E  u.  F >. is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting in two edges between two vertices), analogous to umgraun 23409. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
Hypotheses
Ref Expression
uhgraun.e  |-  ( ph  ->  E  Fn  A )
uhgraun.f  |-  ( ph  ->  F  Fn  B )
uhgraun.i  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
uhgraun.ge  |-  ( ph  ->  V UHGrph  E )
uhgraun.gf  |-  ( ph  ->  V UHGrph  F )
Assertion
Ref Expression
uhgraun  |-  ( ph  ->  V UHGrph  ( E  u.  F ) )

Proof of Theorem uhgraun
StepHypRef Expression
1 uhgraun.ge . . . . . 6  |-  ( ph  ->  V UHGrph  E )
2 uhgraf 23385 . . . . . 6  |-  ( V UHGrph  E  ->  E : dom  E --> ( ~P V  \  { (/) } ) )
31, 2syl 16 . . . . 5  |-  ( ph  ->  E : dom  E --> ( ~P V  \  { (/)
} ) )
4 uhgraun.e . . . . 5  |-  ( ph  ->  E  Fn  A )
5 fndm 5613 . . . . . . 7  |-  ( E  Fn  A  ->  dom  E  =  A )
65feq2d 5650 . . . . . 6  |-  ( E  Fn  A  ->  ( E : dom  E --> ( ~P V  \  { (/) } )  <->  E : A --> ( ~P V  \  { (/) } ) ) )
76biimpac 486 . . . . 5  |-  ( ( E : dom  E --> ( ~P V  \  { (/)
} )  /\  E  Fn  A )  ->  E : A --> ( ~P V  \  { (/) } ) )
83, 4, 7syl2anc 661 . . . 4  |-  ( ph  ->  E : A --> ( ~P V  \  { (/) } ) )
9 uhgraun.gf . . . . . 6  |-  ( ph  ->  V UHGrph  F )
10 uhgraf 23385 . . . . . 6  |-  ( V UHGrph  F  ->  F : dom  F --> ( ~P V  \  { (/) } ) )
119, 10syl 16 . . . . 5  |-  ( ph  ->  F : dom  F --> ( ~P V  \  { (/)
} ) )
12 uhgraun.f . . . . 5  |-  ( ph  ->  F  Fn  B )
13 fndm 5613 . . . . . . 7  |-  ( F  Fn  B  ->  dom  F  =  B )
1413feq2d 5650 . . . . . 6  |-  ( F  Fn  B  ->  ( F : dom  F --> ( ~P V  \  { (/) } )  <->  F : B --> ( ~P V  \  { (/) } ) ) )
1514biimpac 486 . . . . 5  |-  ( ( F : dom  F --> ( ~P V  \  { (/)
} )  /\  F  Fn  B )  ->  F : B --> ( ~P V  \  { (/) } ) )
1611, 12, 15syl2anc 661 . . . 4  |-  ( ph  ->  F : B --> ( ~P V  \  { (/) } ) )
17 uhgraun.i . . . 4  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
18 fun2 5679 . . . 4  |-  ( ( ( E : A --> ( ~P V  \  { (/)
} )  /\  F : B --> ( ~P V  \  { (/) } ) )  /\  ( A  i^i  B )  =  (/) )  -> 
( E  u.  F
) : ( A  u.  B ) --> ( ~P V  \  { (/)
} ) )
198, 16, 17, 18syl21anc 1218 . . 3  |-  ( ph  ->  ( E  u.  F
) : ( A  u.  B ) --> ( ~P V  \  { (/)
} ) )
20 fdm 5666 . . . . 5  |-  ( ( E  u.  F ) : ( A  u.  B ) --> ( ~P V  \  { (/) } )  ->  dom  ( E  u.  F )  =  ( A  u.  B
) )
2119, 20syl 16 . . . 4  |-  ( ph  ->  dom  ( E  u.  F )  =  ( A  u.  B ) )
2221feq2d 5650 . . 3  |-  ( ph  ->  ( ( E  u.  F ) : dom  ( E  u.  F
) --> ( ~P V  \  { (/) } )  <->  ( E  u.  F ) : ( A  u.  B ) --> ( ~P V  \  { (/) } ) ) )
2319, 22mpbird 232 . 2  |-  ( ph  ->  ( E  u.  F
) : dom  ( E  u.  F ) --> ( ~P V  \  { (/)
} ) )
24 reluhgra 23382 . . . 4  |-  Rel UHGrph
25 releldm 5175 . . . 4  |-  ( ( Rel UHGrph  /\  V UHGrph  E )  ->  V  e.  dom UHGrph  )
2624, 1, 25sylancr 663 . . 3  |-  ( ph  ->  V  e.  dom UHGrph  )
27 uhgrav 23383 . . . . . 6  |-  ( V UHGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2827simprd 463 . . . . 5  |-  ( V UHGrph  E  ->  E  e.  _V )
291, 28syl 16 . . . 4  |-  ( ph  ->  E  e.  _V )
30 uhgrav 23383 . . . . . 6  |-  ( V UHGrph  F  ->  ( V  e. 
_V  /\  F  e.  _V ) )
3130simprd 463 . . . . 5  |-  ( V UHGrph  F  ->  F  e.  _V )
329, 31syl 16 . . . 4  |-  ( ph  ->  F  e.  _V )
33 unexg 6486 . . . 4  |-  ( ( E  e.  _V  /\  F  e.  _V )  ->  ( E  u.  F
)  e.  _V )
3429, 32, 33syl2anc 661 . . 3  |-  ( ph  ->  ( E  u.  F
)  e.  _V )
35 isuhgra 23384 . . 3  |-  ( ( V  e.  dom UHGrph  /\  ( E  u.  F )  e.  _V )  ->  ( V UHGrph  ( E  u.  F
)  <->  ( E  u.  F ) : dom  ( E  u.  F
) --> ( ~P V  \  { (/) } ) ) )
3626, 34, 35syl2anc 661 . 2  |-  ( ph  ->  ( V UHGrph  ( E  u.  F )  <->  ( E  u.  F ) : dom  ( E  u.  F
) --> ( ~P V  \  { (/) } ) ) )
3723, 36mpbird 232 1  |-  ( ph  ->  V UHGrph  ( E  u.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   _Vcvv 3072    \ cdif 3428    u. cun 3429    i^i cin 3430   (/)c0 3740   ~Pcpw 3963   {csn 3980   class class class wbr 4395   dom cdm 4943   Rel wrel 4948    Fn wfn 5516   -->wf 5517   UHGrph cuhg 23380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-fun 5523  df-fn 5524  df-f 5525  df-uhgra 23381
This theorem is referenced by: (None)
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