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Theorem umgraun 23197
Description: If  <. V ,  E >. and  <. V ,  F >. are graphs, then  <. V ,  E  u.  F >. is a graph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.)
Hypotheses
Ref Expression
umgraun.e  |-  ( ph  ->  E  Fn  A )
umgraun.f  |-  ( ph  ->  F  Fn  B )
umgraun.i  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
umgraun.ge  |-  ( ph  ->  V UMGrph  E )
umgraun.gf  |-  ( ph  ->  V UMGrph  F )
Assertion
Ref Expression
umgraun  |-  ( ph  ->  V UMGrph  ( E  u.  F ) )

Proof of Theorem umgraun
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 umgraun.ge . . . . 5  |-  ( ph  ->  V UMGrph  E )
2 umgraun.e . . . . 5  |-  ( ph  ->  E  Fn  A )
3 umgraf 23187 . . . . 5  |-  ( ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
41, 2, 3syl2anc 656 . . . 4  |-  ( ph  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
5 umgraun.gf . . . . 5  |-  ( ph  ->  V UMGrph  F )
6 umgraun.f . . . . 5  |-  ( ph  ->  F  Fn  B )
7 umgraf 23187 . . . . 5  |-  ( ( V UMGrph  F  /\  F  Fn  B )  ->  F : B --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
85, 6, 7syl2anc 656 . . . 4  |-  ( ph  ->  F : B --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
9 umgraun.i . . . 4  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
10 fun2 5573 . . . 4  |-  ( ( ( E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 }  /\  F : B --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )  /\  ( A  i^i  B )  =  (/) )  ->  ( E  u.  F ) : ( A  u.  B
) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
114, 8, 9, 10syl21anc 1212 . . 3  |-  ( ph  ->  ( E  u.  F
) : ( A  u.  B ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
12 fdm 5560 . . . . 5  |-  ( ( E  u.  F ) : ( A  u.  B ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  ->  dom  ( E  u.  F )  =  ( A  u.  B ) )
1311, 12syl 16 . . . 4  |-  ( ph  ->  dom  ( E  u.  F )  =  ( A  u.  B ) )
1413feq2d 5544 . . 3  |-  ( ph  ->  ( ( E  u.  F ) : dom  ( E  u.  F
) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  <->  ( E  u.  F ) : ( A  u.  B ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
1511, 14mpbird 232 . 2  |-  ( ph  ->  ( E  u.  F
) : dom  ( E  u.  F ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
16 relumgra 23183 . . . 4  |-  Rel UMGrph
17 releldm 5068 . . . 4  |-  ( ( Rel UMGrph  /\  V UMGrph  E )  ->  V  e.  dom UMGrph  )
1816, 1, 17sylancr 658 . . 3  |-  ( ph  ->  V  e.  dom UMGrph  )
1916brrelex2i 4876 . . . . 5  |-  ( V UMGrph  E  ->  E  e.  _V )
201, 19syl 16 . . . 4  |-  ( ph  ->  E  e.  _V )
2116brrelex2i 4876 . . . . 5  |-  ( V UMGrph  F  ->  F  e.  _V )
225, 21syl 16 . . . 4  |-  ( ph  ->  F  e.  _V )
23 unexg 6380 . . . 4  |-  ( ( E  e.  _V  /\  F  e.  _V )  ->  ( E  u.  F
)  e.  _V )
2420, 22, 23syl2anc 656 . . 3  |-  ( ph  ->  ( E  u.  F
)  e.  _V )
25 isumgra 23184 . . 3  |-  ( ( V  e.  dom UMGrph  /\  ( E  u.  F )  e.  _V )  ->  ( V UMGrph  ( E  u.  F
)  <->  ( E  u.  F ) : dom  ( E  u.  F
) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
2618, 24, 25syl2anc 656 . 2  |-  ( ph  ->  ( V UMGrph  ( E  u.  F )  <->  ( E  u.  F ) : dom  ( E  u.  F
) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
2715, 26mpbird 232 1  |-  ( ph  ->  V UMGrph  ( E  u.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1364    e. wcel 1761   {crab 2717   _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   ` cfv 5415    <_ cle 9415   2c2 10367   #chash 12099   UMGrph cumg 23181
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-umgra 23182
This theorem is referenced by:  uslgraun  23228  eupap1  23532
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