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Theorem umgrale 23873
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 23870 . . . 4  |-  ( ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
2 ffvelrn 5663 . . . 4  |-  ( ( E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
31, 2sylan 457 . . 3  |-  ( ( ( V UMGrph  E  /\  E  Fn  A )  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
433impa 1146 . 2  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
5 fveq2 5525 . . . . 5  |-  ( x  =  ( E `  F )  ->  ( # `
 x )  =  ( # `  ( E `  F )
) )
65breq1d 4033 . . . 4  |-  ( x  =  ( E `  F )  ->  (
( # `  x )  <_  2  <->  ( # `  ( E `  F )
)  <_  2 ) )
76elrab 2923 . . 3  |-  ( ( E `  F )  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  <->  ( ( E `  F
)  e.  ( ~P V  \  { (/) } )  /\  ( # `  ( E `  F
) )  <_  2
) )
87simprbi 450 . 2  |-  ( ( E `  F )  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  ->  ( # `  ( E `  F )
)  <_  2 )
94, 8syl 15 1  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( # `  ( E `  F )
)  <_  2 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    \ cdif 3149   (/)c0 3455   ~Pcpw 3625   {csn 3640   class class class wbr 4023    Fn wfn 5250   -->wf 5251   ` cfv 5255    <_ cle 8868   2c2 9795   #chash 11337   UMGrph cumg 23860
This theorem is referenced by:  umgrafi  23874  umgraex  23875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-umgra 23863
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