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Theorem umgrale 23260
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 23257 . . . 4  |-  ( ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
21ffvelrnda 5848 . . 3  |-  ( ( ( V UMGrph  E  /\  E  Fn  A )  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
323impa 1182 . 2  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
4 fveq2 5696 . . . . 5  |-  ( x  =  ( E `  F )  ->  ( # `
 x )  =  ( # `  ( E `  F )
) )
54breq1d 4307 . . . 4  |-  ( x  =  ( E `  F )  ->  (
( # `  x )  <_  2  <->  ( # `  ( E `  F )
)  <_  2 ) )
65elrab 3122 . . 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 965    = wceq 1369    e. wcel 1756   {crab 2724    \ cdif 3330   (/)c0 3642   ~Pcpw 3865   {csn 3882   class class class wbr 4297    Fn wfn 5418   ` cfv 5423    <_ cle 9424   2c2 10376   #chash 12108   UMGrph cumg 23251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-umgra 23252
This theorem is referenced by:  umgrafi  23261  umgraex  23262
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