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Theorem umgrale 21339
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 21336 . . . 4  |-  ( ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
21ffvelrnda 5856 . . 3  |-  ( ( ( V UMGrph  E  /\  E  Fn  A )  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
323impa 1148 . 2  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
4 fveq2 5714 . . . . 5  |-  ( x  =  ( E `  F )  ->  ( # `
 x )  =  ( # `  ( E `  F )
) )
54breq1d 4209 . . . 4  |-  ( x  =  ( E `  F )  ->  (
( # `  x )  <_  2  <->  ( # `  ( E `  F )
)  <_  2 ) )
65elrab 3079 . . 3  |-  ( ( E `  F )  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  <->  ( ( E `  F
)  e.  ( ~P V  \  { (/) } )  /\  ( # `  ( E `  F
) )  <_  2
) )
76simprbi 451 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2696    \ cdif 3304   (/)c0 3615   ~Pcpw 3786   {csn 3801   class class class wbr 4199    Fn wfn 5435   ` cfv 5440    <_ cle 9105   2c2 10033   #chash 11601   UMGrph cumg 21330
This theorem is referenced by:  umgrafi  21340  umgraex  21341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pr 4390
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-fv 5448  df-umgra 21331
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