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Theorem umgraun 21355
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 21345 . . . . 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 21345 . . . . 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 5600 . . . 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 5587 . . . . 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 5573 . . 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 21341 . . . 4  |-  Rel UMGrph
17 releldm 5094 . . . 4  |-  ( ( Rel UMGrph  /\  V UMGrph  E )  ->  V  e.  dom UMGrph  )
1816, 1, 17sylancr 645 . . 3  |-  ( ph  ->  V  e.  dom UMGrph  )
1916brrelex2i 4911 . . . . 5  |-  ( V UMGrph  E  ->  E  e.  _V )
201, 19syl 16 . . . 4  |-  ( ph  ->  E  e.  _V )
2116brrelex2i 4911 . . . . 5  |-  ( V UMGrph  F  ->  F  e.  _V )
225, 21syl 16 . . . 4  |-  ( ph  ->  F  e.  _V )
23 unexg 4702 . . . 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 21342 . . 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 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311   (/)c0 3620   ~Pcpw 3791   {csn 3806   class class class wbr 4204   dom cdm 4870   Rel wrel 4875    Fn wfn 5441   -->wf 5442   ` cfv 5446    <_ cle 9113   2c2 10041   #chash 11610   UMGrph cumg 21339
This theorem is referenced by:  uslgraun  21386  eupap1  21690
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450  df-umgra 21340
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