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Theorem umgraun 23397
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 23387 . . . . 5  |-  ( ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
41, 2, 3syl2anc 661 . . . 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 23387 . . . . 5  |-  ( ( V UMGrph  F  /\  F  Fn  B )  ->  F : B --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
85, 6, 7syl2anc 661 . . . 4  |-  ( ph  ->  F : B --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
9 umgraun.i . . . 4  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
10 fun2 5674 . . . 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 1218 . . 3  |-  ( ph  ->  ( E  u.  F
) : ( A  u.  B ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
12 fdm 5661 . . . . 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 5645 . . 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 23383 . . . 4  |-  Rel UMGrph
17 releldm 5170 . . . 4  |-  ( ( Rel UMGrph  /\  V UMGrph  E )  ->  V  e.  dom UMGrph  )
1816, 1, 17sylancr 663 . . 3  |-  ( ph  ->  V  e.  dom UMGrph  )
1916brrelex2i 4978 . . . . 5  |-  ( V UMGrph  E  ->  E  e.  _V )
201, 19syl 16 . . . 4  |-  ( ph  ->  E  e.  _V )
2116brrelex2i 4978 . . . . 5  |-  ( V UMGrph  F  ->  F  e.  _V )
225, 21syl 16 . . . 4  |-  ( ph  ->  F  e.  _V )
23 unexg 6481 . . . 4  |-  ( ( E  e.  _V  /\  F  e.  _V )  ->  ( E  u.  F
)  e.  _V )
2420, 22, 23syl2anc 661 . . 3  |-  ( ph  ->  ( E  u.  F
)  e.  _V )
25 isumgra 23384 . . 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 661 . 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 1370    e. wcel 1758   {crab 2799   _Vcvv 3068    \ cdif 3423    u. cun 3424    i^i cin 3425   (/)c0 3735   ~Pcpw 3958   {csn 3975   class class class wbr 4390   dom cdm 4938   Rel wrel 4943    Fn wfn 5511   -->wf 5512   ` cfv 5516    <_ cle 9520   2c2 10472   #chash 12204   UMGrph cumg 23381
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 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629  ax-un 6472
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-fun 5518  df-fn 5519  df-f 5520  df-umgra 23382
This theorem is referenced by:  uslgraun  23428  eupap1  23732
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