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Theorem umgrass 24275
Description: An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
Assertion
Ref Expression
umgrass  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  C_  V
)

Proof of Theorem umgrass
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssrab2 3334 . . . 4  |-  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  C_  ( ~P V  \  { (/) } )
2 difss 3379 . . . 4  |-  ( ~P V  \  { (/) } )  C_  ~P V
31, 2sstri 3264 . . 3  |-  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  C_ 
~P V
4 umgraf 24274 . . . . 5  |-  ( ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
5 ffvelrn 5743 . . . . 5  |-  ( ( E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
64, 5sylan 457 . . . 4  |-  ( ( ( V UMGrph  E  /\  E  Fn  A )  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
763impa 1146 . . 3  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
83, 7sseldi 3254 . 2  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  e.  ~P V )
9 elpwi 3709 . 2  |-  ( ( E `  F )  e.  ~P V  -> 
( E `  F
)  C_  V )
108, 9syl 15 1  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  C_  V
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1710   {crab 2623    \ cdif 3225    C_ wss 3228   (/)c0 3531   ~Pcpw 3701   {csn 3716   class class class wbr 4102    Fn wfn 5329   -->wf 5330   ` cfv 5334    <_ cle 8955   2c2 9882   #chash 11427   UMGrph cumg 24264
This theorem is referenced by:  umgraex  24279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fv 5342  df-umgra 24267
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