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Theorem umgrale 24144
Description: An edge has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.)
Assertion
Ref Expression
umgrale  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( # `  ( E `  F )
)  <_  2 )

Proof of Theorem umgrale
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 umgraf 24141 . . . 4  |-  ( ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
21ffvelrnda 6032 . . 3  |-  ( ( ( V UMGrph  E  /\  E  Fn  A )  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
323impa 1191 . 2  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
4 fveq2 5872 . . . . 5  |-  ( x  =  ( E `  F )  ->  ( # `
 x )  =  ( # `  ( E `  F )
) )
54breq1d 4463 . . . 4  |-  ( x  =  ( E `  F )  ->  (
( # `  x )  <_  2  <->  ( # `  ( E `  F )
)  <_  2 ) )
65elrab 3266 . . 3  |-  ( ( E `  F )  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  <->  ( ( E `  F
)  e.  ( ~P V  \  { (/) } )  /\  ( # `  ( E `  F
) )  <_  2
) )
76simprbi 464 . 2  |-  ( ( E `  F )  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  ->  ( # `  ( E `  F )
)  <_  2 )
83, 7syl 16 1  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( # `  ( E `  F )
)  <_  2 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2821    \ cdif 3478   (/)c0 3790   ~Pcpw 4016   {csn 4033   class class class wbr 4453    Fn wfn 5589   ` cfv 5594    <_ cle 9641   2c2 10597   #chash 12385   UMGrph cumg 24135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-umgra 24136
This theorem is referenced by:  umgrafi  24145  umgraex  24146
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