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Theorem umgraun 21231
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 21221 . . . . 5  |-  ( ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
41, 2, 3syl2anc 643 . . . 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 21221 . . . . 5  |-  ( ( V UMGrph  F  /\  F  Fn  B )  ->  F : B --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
85, 6, 7syl2anc 643 . . . 4  |-  ( ph  ->  F : B --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
9 umgraun.i . . . 4  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
10 fun2 5549 . . . 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 1183 . . 3  |-  ( ph  ->  ( E  u.  F
) : ( A  u.  B ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
12 fdm 5536 . . . . 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 5522 . . 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 224 . 2  |-  ( ph  ->  ( E  u.  F
) : dom  ( E  u.  F ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
16 relumgra 21217 . . . 4  |-  Rel UMGrph
17 releldm 5043 . . . 4  |-  ( ( Rel UMGrph  /\  V UMGrph  E )  ->  V  e.  dom UMGrph  )
1816, 1, 17sylancr 645 . . 3  |-  ( ph  ->  V  e.  dom UMGrph  )
1916brrelex2i 4860 . . . . 5  |-  ( V UMGrph  E  ->  E  e.  _V )
201, 19syl 16 . . . 4  |-  ( ph  ->  E  e.  _V )
2116brrelex2i 4860 . . . . 5  |-  ( V UMGrph  F  ->  F  e.  _V )
225, 21syl 16 . . . 4  |-  ( ph  ->  F  e.  _V )
23 unexg 4651 . . . 4  |-  ( ( E  e.  _V  /\  F  e.  _V )  ->  ( E  u.  F
)  e.  _V )
2420, 22, 23syl2anc 643 . . 3  |-  ( ph  ->  ( E  u.  F
)  e.  _V )
25 isumgra 21218 . . 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 643 . 2  |-  ( ph  ->  ( V UMGrph  ( E  u.  F )  <->  ( E  u.  F ) : dom  ( E  u.  F
) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
2715, 26mpbird 224 1  |-  ( ph  ->  V UMGrph  ( E  u.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   {crab 2654   _Vcvv 2900    \ cdif 3261    u. cun 3262    i^i cin 3263   (/)c0 3572   ~Pcpw 3743   {csn 3758   class class class wbr 4154   dom cdm 4819   Rel wrel 4824    Fn wfn 5390   -->wf 5391   ` cfv 5395    <_ cle 9055   2c2 9982   #chash 11546   UMGrph cumg 21215
This theorem is referenced by:  uslgraun  21262  eupap1  21547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-fun 5397  df-fn 5398  df-f 5399  df-umgra 21216
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