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Theorem uhgraun 23180
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 23197. (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 23173 . . . . . 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 5507 . . . . . . 7  |-  ( E  Fn  A  ->  dom  E  =  A )
65feq2d 5544 . . . . . 6  |-  ( E  Fn  A  ->  ( E : dom  E --> ( ~P V  \  { (/) } )  <->  E : A --> ( ~P V  \  { (/) } ) ) )
76biimpac 483 . . . . 5  |-  ( ( E : dom  E --> ( ~P V  \  { (/)
} )  /\  E  Fn  A )  ->  E : A --> ( ~P V  \  { (/) } ) )
83, 4, 7syl2anc 656 . . . 4  |-  ( ph  ->  E : A --> ( ~P V  \  { (/) } ) )
9 uhgraun.gf . . . . . 6  |-  ( ph  ->  V UHGrph  F )
10 uhgraf 23173 . . . . . 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 5507 . . . . . . 7  |-  ( F  Fn  B  ->  dom  F  =  B )
1413feq2d 5544 . . . . . 6  |-  ( F  Fn  B  ->  ( F : dom  F --> ( ~P V  \  { (/) } )  <->  F : B --> ( ~P V  \  { (/) } ) ) )
1514biimpac 483 . . . . 5  |-  ( ( F : dom  F --> ( ~P V  \  { (/)
} )  /\  F  Fn  B )  ->  F : B --> ( ~P V  \  { (/) } ) )
1611, 12, 15syl2anc 656 . . . 4  |-  ( ph  ->  F : B --> ( ~P V  \  { (/) } ) )
17 uhgraun.i . . . 4  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
18 fun2 5573 . . . 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 1212 . . 3  |-  ( ph  ->  ( E  u.  F
) : ( A  u.  B ) --> ( ~P V  \  { (/)
} ) )
20 fdm 5560 . . . . 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 5544 . . 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 23170 . . . 4  |-  Rel UHGrph
25 releldm 5068 . . . 4  |-  ( ( Rel UHGrph  /\  V UHGrph  E )  ->  V  e.  dom UHGrph  )
2624, 1, 25sylancr 658 . . 3  |-  ( ph  ->  V  e.  dom UHGrph  )
27 uhgrav 23171 . . . . . 6  |-  ( V UHGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2827simprd 460 . . . . 5  |-  ( V UHGrph  E  ->  E  e.  _V )
291, 28syl 16 . . . 4  |-  ( ph  ->  E  e.  _V )
30 uhgrav 23171 . . . . . 6  |-  ( V UHGrph  F  ->  ( V  e. 
_V  /\  F  e.  _V ) )
3130simprd 460 . . . . 5  |-  ( V UHGrph  F  ->  F  e.  _V )
329, 31syl 16 . . . 4  |-  ( ph  ->  F  e.  _V )
33 unexg 6380 . . . 4  |-  ( ( E  e.  _V  /\  F  e.  _V )  ->  ( E  u.  F
)  e.  _V )
3429, 32, 33syl2anc 656 . . 3  |-  ( ph  ->  ( E  u.  F
)  e.  _V )
35 isuhgra 23172 . . 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 656 . 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 1364    e. wcel 1761   _Vcvv 2970    \ cdif 3322    u. cun 3323    i^i cin 3324   (/)c0 3634   ~Pcpw 3857   {csn 3874   class class class wbr 4289   dom cdm 4836   Rel wrel 4841    Fn wfn 5410   -->wf 5411   UHGrph cuhg 23168
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-fun 5417  df-fn 5418  df-f 5419  df-uhgra 23169
This theorem is referenced by: (None)
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