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Theorem umgraun 24455
Description: The union of two (undirected) multigraphs (with the same vertex set): 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 24445 . . . . 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 24445 . . . . 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 5755 . . . 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 1227 . . 3  |-  ( ph  ->  ( E  u.  F
) : ( A  u.  B ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
12 fdm 5741 . . . . 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 5724 . . 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 24441 . . . 4  |-  Rel UMGrph
17 releldm 5245 . . . 4  |-  ( ( Rel UMGrph  /\  V UMGrph  E )  ->  V  e.  dom UMGrph  )
1816, 1, 17sylancr 663 . . 3  |-  ( ph  ->  V  e.  dom UMGrph  )
1916brrelex2i 5050 . . . . 5  |-  ( V UMGrph  E  ->  E  e.  _V )
201, 19syl 16 . . . 4  |-  ( ph  ->  E  e.  _V )
2116brrelex2i 5050 . . . . 5  |-  ( V UMGrph  F  ->  F  e.  _V )
225, 21syl 16 . . . 4  |-  ( ph  ->  F  e.  _V )
23 unexg 6600 . . . 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 24442 . . 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 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    \ cdif 3468    u. cun 3469    i^i cin 3470   (/)c0 3793   ~Pcpw 4015   {csn 4032   class class class wbr 4456   dom cdm 5008   Rel wrel 5013    Fn wfn 5589   -->wf 5590   ` cfv 5594    <_ cle 9646   2c2 10606   #chash 12408   UMGrph cumg 24439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-fun 5596  df-fn 5597  df-f 5598  df-umgra 24440
This theorem is referenced by:  uslgraun  24501  eupap1  25103
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